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Psychology

PSY201H1

Ashley Waggoner Denton

Fall

Description

University of Toronto - Revision Paper - PSY201 Marcus Lam
Professors:
Ashley Waggoner Denton
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Statistics I [PSY201H1F]
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1. Introduction to Statistics
3. Defining and Measuring Variables
2. Frequency Distributions
3. Central Tendency
4. Variability
5. z-Scores: Location of Scores and Standardized Distributions
6. Probability
7. Probability and Samples: The Distribution of Sample Means
8. Introduction to Hypothesis Testing
6. Research Strategies and Validity
9. Introduction to the t Statistic
10.The t Test for Two Independent Samples
11.The t Test for Two Related Samples
12.The Correlational Research Strategy
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Detailed Contents
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V ocabulary
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Datum:Asingle measurement or observation and is commonly called a score or raw score.
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▯ata: Measurements or observations.
Data Set: Collection of measurements or observations.
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▯heories: Statements about the causal relations between two or more variables.
Variables:Acharacteristic or condition that changes or has different values for different individuals
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Statistics: The science of gaining insight from data through mathematical procedures.
Descriptive Statistics: Statistical procedures used to summarize, organize, and simplify data.
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Inferential Statistics: Statistical techniques that allow us to study samples and then make generalization to the
▯opulations from which they were selected from.
Population:Any entire collection of people, animals, plants or things from which we may collect data.
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Sample: Group of units selected from a larger group (the population). Hope to draw valid conclusions about
▯he larger group.
Parameter:Avalue that describes a population, derived from measurements of those individuals.
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▯tatistic:Avalue that describes a sample, derived from measurements of those individuals.
Sampling Error: The discrepancy, or amount of error, that exists between a sample statistic and the
population parameter.
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Chapter 1: Introduction to Statistics
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1.1 Statistics, Science, and Observations
1.2 Populations and Samples
1.3 Data Structures, Research Methods, and Statistics
1.4 Variables and Measurement
1.5 Statistical Notation
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1. Statistics, Science, and Observations
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Datum:Asingle measurement or observation and is commonly called a score or raw score.
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Data: Measurements or observations.
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Data Set: Collection of measurements or observations.
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Four Canons of Science:
1. Determinism: Universe is orderly, events are meaningful, with systematics causes.
a. Theories: Statements about the causal relations between two or more variables.
b. Variables:Acharacteristic or condition that changes or has different values for different individuals.
3. Parsimony aka. Occam’s Razor: When theories are equally good at explaining observations, then choose the
one with fewest assumptions, or the simpler one.
4. Testability: Confirmable or falsifiable using available research techniques
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Statistics: The science of gaining insight from data through mathematical procedures.
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Descriptive Statistics: Statistical procedures used to summarize, organize, and simplify data.
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Inferential Statistics: Statistical techniques that allow us to study samples and then make generalization to the
populations from which they were selected from.
Population:Any entire collection of people, animals, plants or things from which we may collect data.
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Sample: Group of units selected from a larger group (the population). Hope to draw valid conclusions about
the larger group.
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Parameter:Avalue that describes a population, derived from measurements of those individuals.
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Statistic:Avalue that describes a sample, derived from measurements of those individuals.
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Sampling Error: The discrepancy, or amount of error, that exists between a sample statistic and the
▯opulation parameter.
Two Common Data Structures: (for two variables, method depend on nature of variables and type of claim)
1. One sample, from two scores from everyone
a. Correlational: Observe two variables for scatter plot and relationships.
b. Pre-Post studies: Same variable measured twice for each individual through time.
c. Quasi-Independent Variable: Independent variable used to create different groups of scores in an non-
experimental study.
2. Two samples, with one score from everyone
a. Comparing Non-Equivalent Groups: Compare pre-existing groups on variables of interest because they
cannot be manipulated.
b.causal claims. Consist of manipulating the IV, measuring the DV under random assignment to produce
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i. Independent Variable (IV): this varies on purpose
ii. Dependent Variable (DV) this is the effect
iii.Experimental Condition: Receives experimental treatment
iv.Control Condition: Did not receive experimental treatment or receives placebo. Provides baseline for
comparison.
v. Stay alert for confounds: variables affecting DV that is not IV.
1. Participant variables: eg. Gender
2. Environmental variables eg. Time of day
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How to Control Confounds?
1. RANDOMASSIGNMENT takes away participant variables.
2. MATCHING is used when a variable has a huge effect on DV.
3. Holding them CONSTANT eg. fMRI studies only right handed people
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Limitations of Non-experimental Methods:
1. Lack manipulation of variable
2. Lack random assignment
3. Cannot make causal claims
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Types of Claims:
1. Frequency Claims eg. Numbers
2. Association Claims eg. Relationships between variables
3. Causal claims eg. Something makes something else happen
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Calculations in PSY201:
1. BEDMSAS
a. Brackets
b. Exponents
c. Division
d. Multiplication
e. Summation
f. Addition
g. Subtraction
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Notation:
1. Scores are referred to as X and Y
a. N = number of scores in a population
b. n = number of scores in a sample
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*Use a computational table where the first column is the original data, and every successive column is one step in an
operation
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Chapter 2: Operational Definitions and
Measurements
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2.1
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2. Operational Definitions and Measurements
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Testability:
1. Atheory needs to be testable, falsifiable, have a way to be measured, observed, and proved wrong.
Constructs: Internal attributes or characteristics that cannot be directly observed but are useful for
describing and explaining behavior.
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Conceptual Definition: Provides us with the meaning of the variable.
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Operational Definitions: Provides us with a way to measure it, describes a set of operations, defining the
construct in terms of the resulting measurements.
Dependent Variable: What you are measuring.
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Independent Variable: What you are manipulating.
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2.1 Types of Validity
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Construct Validity: How well does our measurement procedure capture the construct (variable) we are
interested in? Or How well does our operational definition map on to the conceptual definition of the
1. Show construct validity by showing that the results from our measurement procedure are affected by the same
factors that we know affect the variable we are interested in.
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Convergent Validity: Using more than one measurement in order to capture different aspects of the same
construct to see if results converge.
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Divergent Validity: Showing that your measures of the construct you are interested in do not converge (they
diverge) from measures of different constructs.
Concurrent Validity: Do the results from our new measurement procedure correspond with the results from
more well-established measure?
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Face Validity: Does it look like we are measuring what we want to measure?
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Predictive Validity:Are the measures we obtain predictive of behavior (in the manner we would expect,
according to our theories)
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2.2 Reliability
1. Sa. Inter-rater reliability measurement
b. Internal consistency
2. General rule: more is better!
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2.3Accuracy
1. Accuracy refers to the extent to which an experimental measure is free from error.
2. Can only be examined for certain types of measures
3. Two types of error (different types of inaccuracies)
a. Random Error
b. Systematic Error
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2.4 Different Types of Variables
1. Demographics
2. Physical characteristics
3. Individual differences
4. Beliefs and attitudes
5. Performance variables
6. Independent variable
7. Dependent variable
8. Extraneous variables
9. Confounding variables
10.Participant variable
11.Environmental variables
12.Discrete Variable: Consist of separate, indivisible categories
13.Continuous Variable: Divisible into an infinite number of fractional parts
a. You can tell when you are free to decide the degree of precision of the measurement
b. Each measurement category is actually an interval that must be defined by boundaries because:
i. Two children who are 7 years old are probably not the exact same age
ii. Two people who are same height in inches are probably not the exact same height
c. Setting up interval boundaries
i. Real Limits: The boundaries of intervals for scores that are represented on a continuous number line.
The real limit separating two adjacent scores is located exactly halfway between the scores.
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2.5 Scales of Measurement
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Nominal Scale: Categories have different names, but are not related to each other in any systematic way -
there are no quantitative distinctions between observations.
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Ordinal Scale: Categories are organized in an ordered sequence with directional relationship. Sometimes
identified by verbal labels.
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Interval Scale: Measurement with a series of equal intervals with an arbitrary zero point.
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Ratio Scale: Measurement of the absolute magnitude with a series of equal intervals with a meaningful zero
point.
1. It allows us to describe the differences in terms of a ratio which we cannot do with interval scales
a. Someone who is 6 feet tall is twice as tall as someone who is 3 feet tall
b. But 20 degrees is not twice as warm as 10 degrees
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Likert Scale: Participants indicate their degree of agreement or disagreement with a list of statements about
the object.
1. Responses correspond to a numeric rating.
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2. Responses to multiple items are generally added or averaged to provide a single score.
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Usually, we will be dealing with data that ha been measured along an interval or ratio scale and comparing these
measures along different groups of nominal or ordinal categories.
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2.6 BadAspects of Measurement:
1. Range Effects: Unable to distinguish differences between data
a. Ceiling effects: all the data are at the highly extreme
b. Floor effects: all the data are at the low extreme
2. Artifacts:
a. Experimenter bias: experimenter wants to see something happen, conformation bias
b. Participant reactivity: experimenter’s attitude changing the participants performance
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Chapter 3: Frequency Distributions
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3. Frequency Distributions
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Frequency Distribution: How many scores are located in each category of measurement.
1. May be structured as either table or a graph with two columns:
a. SET OF CATEGORIES (X) in original measurement scale
b. FREQUENCY (f), or number of individuals in each category.
c. Note: N = Σf
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*Recall N = population, n = sample
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Theories (x) Frequency (f) p %
1 4 0.2 20
2 2 0.1 10
3 1 0.05 5
4 1 0.05 5
5 5 0.25 25
6 4 0.2 20
7 3 0.15 15
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Proportion aka. Relative Frequency: Fraction of the total group that is associated with each score.
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Percentage: Simply the proportion multiples by 100.
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Methods of Finding the Sum:
1. Take the scores and all them all up
2. Multiply each X value by its f, then add the products.
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X Frequency ΣX X^2 ΣX2 p %
(f)
6 3 18 36 108 0.15 15
5 4 20 25 100 0.2 20
4 6 24 16 96 0.3 30
3 1 3 9 9 0.05 5
2 2 4 4 8 0.1 10
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X Frequency ΣX X^2 ΣX 2 p %
(f)
1 4 4 1 4 0.2 20
20 73
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Percentile Rank: Percentage of individuals in the distribution with scores equal or less than that particular
value.
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Percentile: Score identified by its percentile rank.
1. Eg. “80 percentile” indicates that 80% of the scores in the distribution are equal to or lower
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Cumulative Frequencies (cf): Number of scores in or below each category in the frequency table (accumulates
as you move up the table)
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Cumulative Percentages (c%): The percent in or below each category (Accumulates as you move up the table)
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Interpolation: Method for estimating intermediate values.
Steps:
1. Determine the WIDTH of the interval on both scales:
a. Percentages: 60-26.67 = 33.33
b. Scores: 8.5-7.5 = 1 *not 8.5-6.5 because percentages
correspond to UPPER real limit of interval as we consider
time as a continuous variable*
60%
26.67%
7.5 8.5
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2. Position (of percentile rank) divide the whole interval
a. = 10/33.33 = 0.3
b. One particular percentile (50 ) is located 10 away from the top of the interval (60 )
c. On the score scale, approximately one third down is 0.3(1) = 0.3
3. Real upper limit subtract fraction
a. he top of the interval is 8.5 so 0.3 points down is 8.2.
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Grouped Frequency Distribution Table:
1. Used when it becomes unruly to create frequency tables of individual scores there are
many possible values.
2. So we group the scores into intervals
3. We decide magnitude of interval from the balance between simplicity and meaning
4. Steps: eg. (N = 20) 16, 25, 22, 25, 20, 12, 14, 17, 17, 29, 21, 35, 13, 15, 25, 24, 19, 14,
31, 2
a. How many rows would a regular table need?
i. Highest Score: 35
ii. Lowest Score 12
iii.35-12+1 = 24 rows -> too many!
b. What is a good interval width?
i. Trial and error: Maybe 2? = 12 intervals Maybe 5? = 5 intervals (too few)
c. Identify the intervals
i. Lowest score = 12
ii. Bottom interval 12-13 (why not 11-12?) due to real limits being 11.5-13.5
d. Complete Table:
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Practice: *Interval magnitude should be multiple of lower limit
1. How many rows would a regular table need?
a. Highest Score: 405
b. Lowest Score: 25
c. 405-25+1 = 381 rows -> too many!
2. What is a good interval width?
d. Trial and error: Maybe 10? = 39 intervals Maybe 20? = 20
intervals Maybe 40? = 10 intervals
e. Identify the intervals
i. Lowest score = 25
ii. Bottom interval 20-40 20-59
f. Complete Table:
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X f
1-40 1
41-80 4
81-120 0
121-160 2
161-200 4
201-240 1
241-280 1
281-320 1
321-360 2
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X f
361-400 3
401-440 1
20
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Stem and Leaf Displays:
1.Advantage: Over the group frequency table, we know the frequency of every
▯ndividual score.
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Frequency Distribution Graphs:
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1.misleading graphs. Squished graphs will be misleading.graph’s width because it is to prevent people from producing
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Misleading Graphs:
1. Biggest (or perhaps easiest) trick is to simply not have hte y-axis start at zero
▯. Scale is adjusted to make differences seem larger or smaller than they really are.
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Histograms:
1. For interval or ratio data
2. Heigh of the bars corresponds to the frequency
for that category
3. No space between bars
4. Can also be used for grouped frequency data
where the width of bars extends to the real limits
of the intervals.
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Modified Histograms:
1. Stacked blocks, no y-axis
2. Each block represents one individual (one score)
3. Used throughout textbook to show sample data.
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Polygons:
1. Also for interval or ratio data
2. Vertical position of dot corresponds to frequency of that category.
3. Start and end but extending the line to zero about one interval across 6-> 7, 1-> 0
4. For a grouped distribution, dots are positioned above the midpoint of the class interval
5. Demonstrates the average of the highest and lowest scores in the interval
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Bar Graph:
1. For Ordinal or nominal data
2. Similar to histogram but with space between each bar which emphasizes that
they are distinct categories (nominal scale) or that the categories may not be
the same size (ordinal scale)
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Population Distributions:
1. Relative Frequencies
a. Rare to know absolute frequencies for large populations
b. Easier to obtain relative frequencies, and this can be represented in a bar graph
i. Eg. That women are almost twice as likely as men to suffer from depression -> don’t need exact numbers
to show this
2. Smooth Curves
a. Histograms and polygons become smoothes out (Rather than jagged) to reflect the fact that we’re showing
relative changes from one score to the next, not actual frequencies
i. E.g IW scores 0> normal distribution
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1. Three characteristics that completely describe any distribution:
a. Shape
i. Symmetrical: each side of the distribution is a mirror image of the other
ii. Skewed: Scores are piled up toward one end of the scale and taper off gradually at the other end. Type of
skew: look at the tail (not the “pile”)!
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b. Central Tendency
c. Variability
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Chapter 4: Central Tendency & Variability
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4. Central Tendency & Variability
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Recall* There are three characteristics that completely describe any distribution
1. Shape
2. Central Tendency
3. Variability
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4.1 Central Tendency
Central Tendency:Astatistical measure to determine a single score that defines the center of a distribution.
Goal: Find a single most representative score of your data
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There are three methods for measuring central tendency:And each has been developed to work best in specific
situations
1. The Mean
2. The Median
3. The Mode
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4.1.1 The Mean
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Mean: The sum of the scores (∑X) divided by the total number of scores (N or n).
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Population Mean: µ (“mu”), µ = ∑X / N
▯. eg. IQ scores µ = 100
Sample Mean: M or ▯ “X-Bar”, M = ∑X / n
1. can be used as an estimate of the population mean
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Alternative Definitions:
1. Can think of mean as an EQUAL DISTRIBUTION
a. The mean is the amount each individual would receive if the total was divided equally among all individuals
in the distribution
b. What if you were given the mean (rather than individual scores) and to find the total?
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M = 3.5, n = 4
M = ∑X / n
M*n = ∑X
14 = ∑X
2. Can think of mean as a BALANCE POINT
a. Think of the distance of each score from the mean
b. The total distance below the mean is the same as the
total distance above the mean
c. If we add a block on the right, we need to shift the
middle to the right!
d. By multiplying/dividing scores by a constant, we can
do the same to mean.
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M = 3.5
Score: 4, distance = 0.5 above
Score: 3, distance = 0.5 below
Score: 6, distance = 2.5 above
Score: 1, distance = 2.5 below
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The Weighted Mean
1. What if we have two sets of scores, and we want to find the overall mean?
a. Sample 1 data: 3, 5, 12, 8, 9, 7, 10
c. **We cant just find mean of sample one, and then sample two because we have more data in sample 1, so we
want to give sample 1 more weight to reflect more participants.
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Method:
1. Find ∑X for each sample, then add for the combined ∑X.
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2. Find n for each sample, then add for the combined n.
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3. Divide the combined sum of the scores (∑Xcombined) by the combined number of scores (ncombined)
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Weighted Mean Example One: One section of a course with 300 students scored an average of 120 points on the
test.Another section with 175 students scored an average of 110 points on the test. What is the average test score?
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Total n = 475
120*(300/475) + 110*(175/475) = 116.32
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We are giving special weight on first section because there are a lot more students.
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Weighted Mean Example Two: Given the information below, find the weighted mean of the combined samples:
1. Sample 1 has 14 individuals and the total score of the sample is 126
2. Sample 2 scores: 12, 5, 18, 2, 4, 0, 8
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Sample 2 total score = 49
Total score = 175
Total n = 14 + 7 = 21
Weighted Mean = 175/21 = 8.33
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Finding the Mean from a Frequency Distribution Table
1. M = ∑X / n
3. We can calculate n from frequency distribution table by: n = ∑fn table
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Example: In this example, the mean is also the midpoint of the distribution of scores.
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1. Median: Midpoint of the list from smallest to largest.
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Example: 1,3,4,6
No middle score, so we find average of middle two scores: (3+4)/2 = 3.5
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What is the mean now? 8.2
What is the median? 4
Which is most representative of our data? Mehhh But, **Mean is very influenced by extreme score, pulls the mean
towards it. Median is much less affected by extreme scores.**
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▯he median divides the scores into two groups of equal size, dividing the area of the graph exactly in half:
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Precise Median for a Continuous Variable:
1. However, a CONTINUOUS VARIABLE can be infinitely divided.
2. The PRECISE MEIDAN is located in the interval defined by the real limits of the value
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It is necessary to determine the fraction of the interval needed to divide the distribution exactly in half:
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Example:
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**Note: lines of boxes are lower and upper limits
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Precise Median & Percentile
Notice: The precise median is identical to the 50 percentile for a distribution!
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Precise Median Example: Calculate the precise median for the following group of scores: 3, 8, 2, 5, 4, 9, 6, 4, 13
1. List in order: 2, 3, 4, 4, 5, 6, 8, 9, 13
2. Find the middle score: 5
3. Real limits: 4.5 and 5.5
4. Number of scores to reach 50%? 0.5
5. Number of scores in interval? 1
6. Fraction = ½ or 0.5
7. Precise Median = 4.5 + 0.5 = 5
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The Median and Stem & Leaf Plots
1. Calculate the median for the following group of scores: 33, 28, 22, 25, 24, 19, 26, 14, 13
2. Always do it like the LEFT example! Smallest to largest!
3. Median = 24 when we cross out the extremes and work our way to the middle.
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4.1.3 The Mode
1. The mode is the most common observation among a group of scores. In a frequency distribution, the mode is
simply the score or category that has the GREATEST FREQUENCY.
2. It can be used with any scale of measurement
3. Corresponds to an actual score in the data
4. It is possible to have more than one mode:
a. 2 modes is BIMODAL
b. More than 2 modes is MULTIMODAL
c. Even if data suggests a definite mode, if there is another peak like the data below. It may make sense to talk
about a MAJOR MODE and a MINOR MODE.
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d. If we have a distribution with a lot of modes, there really is no meaningful mode.
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4.1.4 Which Measure of Central Tendency to Use?
1. Each has been developed to work best in certain situations
2. The MEAN is appropriate to use except in the cases of:
a. Extreme scores
b. Skewed distributions
c. Ordinal or nominal scales
d. Undetermined values
e. Open ended distributions
f. In these cases, the MEDIAN is a more appropriate measure of central tendency
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1.can be to infiniteis open ended!It
2. The median is good because it is looking at the number of scores - 50% of families fall below/is above this
income value
3. The mean income is good because it actually tell us how extreme the open ended category is. It demonstrates how
skewed the data is.
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Recall* our discussion of LIKERT SCALE construction
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Can only find a mean if we are treating it as an interval scale -> should only do this if the response options are
SYMMETRIC and EQUIDISTANT.
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The MODE is appropriate to use in cases of:
1. Discrete variables - silly to use fractional points
2. Nominal scales
4. As a supplementary measure
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Symmetrical Distributions
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Skewed Distributions
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Imagine that you are diagnosed with a rare form of cancer, and you read that: “Mesothelioma is incurable, with a
median mortality of only eight months after discover.”
1. 50% of people die before eight months, 50% of people die after eight months
2. Now we need to consider the variability
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Graphing Means & Medians
1. Means most often presented graphically (sometimes medians, never modes)
2. We are often interested in comparing means (or medians) between different groups/conditions
3. Type of graph (Eg. Bar, histogram) depends on the scale of measurement used for the independent variable (Eg.
Nominal or interval) -> x-axis
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4.2 Variability
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Variability:Aquantitative measure of the differences between the scores in a distribution and describes the
degree to which the scores are spread out or clustered together.
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Why do we care?
1. Describes the distribution
a. How much distance between one score and another?
b. How much distance between one score and the mean?
2. It measures how well an individual score (or group of scores) REPRESENTS the entire distribution
a. Inferential Statistics: Variability tells us how much error to expect if we are using a sample to represent a
population
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b. How well would each of these individual scores describe/represent the distribution they are from?
c. How much error to expect?
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The Range
1. Easiest way to determine how spread out the scores are.
2. Range = Xmax - Xmin
3. For continuous variables: Use upper real limit (URL) of Xmax and lower real limit (LRL) of Xmin
5. Sample: 1,25,26,27,28,27,26,25,89 scores in the data
6. Range = 88, but most of the scores actually cluster closely together!
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4.2.1 Variance & Standard Deviation
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Variance:
2. Variance: The average squared distance from the mean.
3. SQUARED DISTANCE is not that useful or easy to understand, which is why we use the standard deviation
(square root of the variance)
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Standard Deviation:Ameasure of the standard distance of a score from the mean.
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Deviation: Distance from the mean.
**Deviation Score = X - µ
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Mean Deviation: Always 0.
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Solution? Get rid of negative numbers by squaring the deviation scores, then find
the mean… this is the VARIANCE!
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square root and this is the STANDARD DEVIATION!t generally want the mean squared deviation. So.. We take the
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Standard Deviation = √Variance
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**Importantly, the calculations differ from populations versus samples, but the concept is the same
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Population Variance & Standard Deviation
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SS aka. Sum of Squares = Sum of squared deviation scores
N = Number of scores in population
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How do we calculate SS?
Deviation score = X - µ
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Definitional Formula: SS = ∑ (X - µ)
Note: This formula is easy to understand, and easy to use when the mean is a whole number. But can lead to errors
when dealing with decimals.
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Example: Compute the SS for the following set of scores: 2, 3, 3, 0
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Computational Formula:
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Remember: The reason we’re interested in SS is to calculate variance and standard deviation.
Population Variance = σ = SS/N
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Population Standard Deviation = σ = √σ = √(SS/N)
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For our example:
SS = 6 N = 4
σ = 6/4 = 1.5 σ = √1.5 = 1.225
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Graphical Representation of Population Mean and Standard Deviation
1. About 68% of scores are going to fall between 1 standard deviation of our population mean.
2. 95% of scores are going to fall between 2 standard deviations
3. 98% of scores are going to fall between 3 standard deviations
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Sample Variance & Standard Deviation:
1. Remember: Goal of inferential statistics is to
draw conclusions about a population based on
information we collect from a sample
2. Samples tend to be less variable than their
populations
3. When you only have a sample to work with,
you can use the sample variance and sample
standard deviation to estimate the population
variance and population standard deviation
4. For this reason, sample variance and standard
deviation are often referred to as estimated
population variance and estimated population
standard deviation.
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Computing sample variance and standard
deviation in the same way as we do for
populations would result in a BIASED estimate of
the population values. Specifically, we would tend
to underestimate the variability in the population.
Because the bias is consistent, it can be corrected!
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Find the SS for a sample:
1. Same as population, except different notation:
a. Find the deviation for each score: X - M
b. Square each deviation = (X - M)2
c. Add the squared deviations: SS = ∑ (X - M) 2
d. Computational Formula:
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Because sample variability tends to underestimate population variability, so we divide by n - 1 instead of n
1. Dividing by a smaller number to produce a larger (unbiased) estimate of population variance.
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Predicting Sample Standard Deviation
1. Largest distance from mean = 4 points
2. Smallest distance from mean = 1 points
3. Therefore, the standard distance from the mean should be between 1 and 4 points (Eg. Maybe 2.5)
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computational formulas for checking answers.iation using the data presented here and use both the definitional and
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N = 7 ∑X = 35
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M = ∑X / n = 35 / 7 = 5
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SS = = ∑ (X - M) = 36
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∑X = 211
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SS = ∑X - (∑X) / n = 211 - 35 / 7 = 36
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S = SS / (n-1) = 36/6 = 6
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▯ = √6 = 2.45
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Degrees of Freedom
1. M must be calculated before we compute anything else (Eg. SS)
2.“free”lating the value of M places a restriction on the variability of scores in our sample.All of the scores are not
a. Specifically, our final score is always going to be restricted to a particular value
3. Degrees of Freedom (df): The number of scores in a sample that are free to vary.
a. For a sample variance, df = n - 1
4. The number of DEGREES of FREEDOM is the number of independent observations in a sample of data that are
available to estimate a parameter of the population from which that sample is drawn
a. We often use a sample variance (a statistic) to estimate population variance ( a parameter)
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▯ote: It is restricted because we know the mean going in to things.
1. This is a sample, we don’t know the mean of population that it is from. So we use M as an estimate of µ. When
calculating M, all of the scores that we are using to estimate this parameter are free to vary (there are no
restrictions).
2. Next, we want to use the sample variance as an estimate of the population variance. But because we had to
calculate M first, all of our scores are no longer free to vary when calculating our estimate of variance.
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Unbiased Statistics:
On average, the sample mean and sample variance (using n - 1) provide an unbiased estimate of the population mean
and population variance. This is going to be important later on! (inferential statistics).
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Harder to detect differences because…?
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Think about it…
1. If you add a constant to each score in your data, will this change the standard deviation? Why or why not? No,
difference from mean will be the same.
2. If you multiple each score in your data set by a constant, will this change the standard deviation? Why or why
not? Yes, it amplifies the difference from the mean.
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5. z-Scores: Location of Scores & Standardized
Distributions
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Purpose: How to compare how good an apple to is to how good an orange is?
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5.1 z-Scores: Location of Scores
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Standard score aka. Z-Score. Knowing about the distribution (eg. Mean, standard deviation) helps us make meaning
out of raw scores.
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One purpose of transforming scores form raw scores into z-scores is because z-scores tell us exactly where the score
is LOCATED within the distribution (Above, below the mean)
Az-Score has two components:
1. Sign (+/-): Indicates DIRECTION from the mean
2. Number: Indicates the DISTANCE from the mean.
3. Z-scores transform scores into standard deviation
units:
a. Az-score of 1.00 indicates a score that is 1
standard deviationABOVE the mean
b. Az-score of -1.50 indicates a score that is 1.5
standard deviations BELOW the mean.
c. Az-score of 0.00 is the mean.
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Characteristics of the Standard Normal Curve:
1. The total area under the normal curve is equal to 1
2. The distribution is mounded and symmetrical and
extends infinitely in both directions along the x-axis. Never touches X-axis
3. The distribution has a mean of 0 and standard deviation of 1 th
4. The mean divides the area in half. 50% of the area is above and 50% is below (mean = median = 50 percentile)
5. Nearly all area is between z = -3.00 and z = +3.00
Example One:
1. Exam score = 81
2. First distribution: mean = 95, SD = 8
3. Z = (81-95) / 8 = -1.75
4. We know mean SD is 8, difference between
81 and 94 is 14, so z-score should be just
less than 3 SD below the mean.
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Example Two: Going the other way round determining raw score from z-score
1. Mean = 95, SD = 8
2. X when z-score is 2.00?
3. X = µ + z(SD) = 95 + 2(8) = 111
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5.2 z-Scores: Standardized Distributions
By STANDARDIZING an entire distribution (transforming all scores into z-scores), we are able to make
comparisons to other standardized distributions.
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If the data being compared come from normal distributions, we can transform the data so that the two sets are
equivalent, we can compare apple to oranges.
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Example: Doug vs. Linda
1. Doug:
a. Annual sales = $56 (k)
b. Avg. Cutomer ratins = 6.5
2. Linda:
a. Annual sales = $42 (k)
b. Avg cutomer rating = 8.2
c. Sales Mean is 46k, SD = 5
d. Customer rating mean is 7.2, SD is 1.5
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Az-score distribution will always have a MEAN of ZERO.
Az-score distribution will always have a STANDARD DEVIATION of ONE.
Transforming a distribution into a distribution of z-scores will NOT change the SHAPE of the distribution, scores
are being RELABELED not repositioned.
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If we want to find overall, we can average out z-scores of sales and customer satisfaction. doug = 0.765, linda =
0.585
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Other Standardized Distributions:
1. After transforming a distribution of raw scores into a z-score distribution, sometimes it is desirable to create new
values for mean and standard deviation. Usually to avoid negative values and decimals
2. For example most IQ tests are standardized so they have a mean of 100 and a standard deviation of 15. Question:
An school administrator wants to standardize an intelligence test so that the new standardized distribution has a
mean of mean = 100 and a SD of 15. The original distribution of the test was mean = 89 and SD of 12. Johns
original score was 95. Whats John’s new, standardized score?
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Step 1: Transform raw scores into z-scores
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Step 2: Change each z-score into a X-value in the new standardized distribution that has a mean of mean = 100 and
standard deviation of SD = 15
1. John z-score
2. Means johns score is half a standard deviation above the mean, so 0.5*15 = 7.5
3. If the mean is 100, SD is 15, then john’s standard score is 100 + 7.5 = 107.5
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Computing z-scores for a sample
1. It is done in the exact same way, just using the sample mean and sample standard deviation
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Using z-scores to determine how “typical” a score is:
1. Interpretation of research results depends on determining if the (treated) sample is noticeably different from the
2. Once technique for defining noticeably different uses z-scores.
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Isaiah old: 71
Isaiah new: 80.147
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Maria old: 72.94
Maria new: 83
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Aplia:
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Purpose of transforming a set of raw scores into a set of z-scores:
1. Make distribution with mean of 0 and SD of 1
2. Be able to describe location of raw score in the distribution
▯. Compare scores form two different distributions
Z-score is a standardized score, standardized score is not necessarily a z-score
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Parameter: Characteristic of a population, usually a numerical value.
Statistic: Characteristic of a sample, usually a numerical value.
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6. Probability & Distribution of Sample Means
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6.1 Introduction to Probability
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ThreeAxioms:
1. The probability of any event is between 0 (no chance of occurring) and 1 (100% chance of occurring)
2. The sum of all possible outcomes equals to 1
a. Outcome: Individual thing that can happen.
b. Event:Aset of outcomes.
3. If eventAis a subset of event B, then the probability ofAmust be less than the probability of B.
a. Eg. Roll a die, prob(4) = ⅙, prob(even) = ½
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Types of Probability
1. Classical:Afinite number of equally likely outcomes. (eg. Flipping a coin, rolling die)
2. Fa. Used in most statistical inference procedures likely something is. (eg. How loaded is the die?)
3. Subjective: Measure of belief that a particular event will occur.
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6.2 Role of Probability in Statistics
If we know the makeup of a population, we can determine the probability of obtaining specific samples.
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Population 1: Regular M&Ms (15% red, 15% green, 85% other)
Population 2: Holiday M&Ms (50% red, 50% green)
Eg. We would expect more red and green from population 2. If we select one green M&M (n = 1), we will guess that
it came from population 2.
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Inferential Statistics
1. For a situation in which several different outcomes are possible, the probability for any specific outcome is
defines as a fraction or a proportion of all the possible outcomes.
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Random Sampling
1. Each individual in the population has an equal chance of being selected.
2. Probabilities must stay constant from one selection to the next. N must be constant; must return each individual to
the population before you make the next selection.
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Probability & Frequency Distributions
1. Proportions and probabilities are equivalent
2. Whenever a population is presented in a frequency distribution graph, it is possible to represent probabilities as
proportions of the graph.
3. Probability can be described as proportions of area contained in each section of a normal distribution.
4. Z-scores are used to identify sections. Eg. -1 to 1 contains 68% of all possibilities.
CHECK POINT! Probability predicts what a sample is likely to be like, it doe not predict population.
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Probabilities & Proportions
1. They are two different ways of asking the same question
2. Example 1: Population of adult heights is normal with a mean of µ = 68 inches and a standard deviation of σ = 6
inches. What is the probability of randomly selecting an individual from this population who is taller than 80
inches?
a. In probability, this means p(X > 80) = ?
b. SKETCH & SHADE
c. Set of all possible heights is the entire area under the graph
d. All heights greater than 80 is area to the right of 80
e. Step 1: Find z-Score. z = (80-68)/6 = 2
f. Step 2: Use Unit Normal Table. p(X > 80) becomes p(z > 2) = 2.28%
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6.3 Using the Unit Normal Table
1. Z-Scores: Tells you where the vertical line separating the body and the tail is located.
a. Positive z-Scores: Body on the left, tail on the right.
b. Negative z-Scores: Body on the right, tail on the left.
c. Zero z-Score: Body and Tail are equal.
2. Table does not list negative z-Score values, but even if a z-Score is negative, proportions are always positive!
3. Notice that “Proportion of distribution to the left of X” is the same as percentile rank!
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CHECK POINT! If you know the probability, you can know the z-Score!
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6.4 Finding Probabilities for Scores
Practice: Aconsumer survey indicates that the average household spends µ = $155 on groceries each week. The
distribution of spending amounts is approximately normal with a standard deviation of σ = $25. Based on this
distribution:
1. What proportion of the population spends more than $175 per week on groceries?
a. z = (170-155)/25 = 0.6
b. Convert using table from the tail = 0.2119 = 21.19%
2. What is the probability of randomly selecting a family that spends less than $100 per week on groceries?
a. Z = (100-155)/25 = -2.2
b. Convert using table from the tail
3. How much money do you need to spend on groceries each week to be in the top 20% of the distribution? (hint:
need to work backwards!)
a. Look at z-Score with tail of 0.2
b. Convert z-Score into z using x = µ + zσ = 176
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Probabilities/Proportions Located Between Two Scores
1. We use column D (proportion between mean and z) in the table
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Binomial Distribution
1. Bivariate or Dichotomous Variable:Any variable with only two possible outcomes (Aor B), not necessarily
50/50.
2. Binomial Probability Distribution: Distribution of probabilities for each outcome of a bivariate variable.
Eg. If I flip a coin 4 times, what are the probabilities associated with each possible outcome (ie. Getting 0 heads, 1
head… 4 heads)
a. p = probability ofA, p(A) eg. Probability of heads
b. q = probability of B, p(B) eg. Probability of tails
c. n = number of observations in the sample eg. Number of coil flips
d. X = number of timesAoccurs in the sample eg. Number of heads obtained
3. As the number of observations increases, the binomial distribution looks increasingly like a normal distribution! If
np and nq are both ≥ 10, it is appropriate to compute probabilities directly from z-scores and the unit normal table.
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The NormalApproximation
1. If np and nq are both ≥ 10
a. µ = np
b. σ = √(npq)
2. We can calculate a z-Score corresponding to each value of X in the distribution. Eg. What is the probability of
obtaining more than 16 heads in 20 coin flips?
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3. We should use the upper real limit of the interval for X. Think of the histogram! So 16.5 instead of 16.
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Inferential Statistics:
Many research situations begin with a population that forms a normal distribution.Arandom sample is elected and
receives a treatment, goal is to evaluate the treatment. Probability is used to decide whether the treated sample is
“noticeably different” from the population.We use sample data to decide between two alternatives:
1. Treatment has no effect
2. Treatment has an effect.
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Extreme values are defined as extreme (5%). So, it is 2.5% in each tail (z = -1.96 and z = 1.96) are scores that are
very unlikely to be obtained from the original population and therefore provide evidence of a treatment effect.
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6.5 What is Distribution of Sample Means
As you know, most research studies involve samples of more than one individual! In these cases, the SAMPLE
MEAN (rather than a single score) is used to make inferences about a population.
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Sampling Error: The natural discrepancy, or amount of error, between a sample statistic and its
corresponding population parameter.
1. Asample does not provide a complete picture of the population because it underestimates variability.
2. Samples differ from each other, so sample means differ from each other
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Distribution of Sample Means: Collection of sample means for all the possible random samples of a particular
size (n) that can be obtained from a population.
1. This is a distribution of statistics rather than a distribution of scores.
2. In order to find the probability of obtaining any specific sample mean, we first need to know all of the possible
sample means.
3. Once we have specified the distribution of sample means, we will be able to find the probability of selecting any
specific sample mean.
4. Constructing the distribution of sample means is only possible in overly simplified situations. Particularly
unrealistically small population and small sample size.
5. In real world situations, we don’t have to actually construct the distribution of sample means because it is
impossible. We use the CENTRAL LIMIT THEOREM to tell us what the distribution of sample means would
look like if we were to actually construct it.
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Central Limit Theorem
For any population with a mean µ and a standard deviation σ, the distribution of sample means for sample size n will
have a mean of µ and standard deviation σ/√n and will approach a normal distribution as n approaches infinity. With
n = 30, distribution is almost perfectly normal.
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Shape: The distribution of sample means is almost perfectly normal if either of the two following conditions is
satisfied:
a. The population from which the samples are selected is normal
b. The number of scores (n) in each sample is relatively large (approx. 30 or more)
This makes sense because we would expect the sample means to cluster around the population mean µ
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Mean: The mean of the distribution of sample means is equal to the population mean µ, and is called the
EXPECTED VALUE OF M. When all of the possible sample means are collected, the average value is identical to µ.
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Variability: The standard deviation for the distribution of sample means, represenMed as σ is called the
STANDARD ERROR OF M. The standard error provides a measure of how much difference is expected from one
sample to another. The standard error provides a measure of how much the distance, on average, is expected between
the sample mean (M) and the population mean (µ) because the overall mean for the distribution of sample means is
equal to µ.
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Standard Error: The average distance between a sample mean and the population mean. It is extremely useful
because it tells us exactly how much error to expect. The size of standard error depends on:
1. Size of Sample (n): Increases, error decreases. LAW OF LARGE NUMBERS
2. Standard Deviation of the population from which the sample is selected: this is the
starting point for standard error. If n = 1, SD = standard error.As sample size
increases, standard error decreases.
Law of Large Numbers: As sample size increases, standard error decreases. We are
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changing the size of the sample we are dealing with. When sample size is increased, the means are going to be more
clustered around the mean.
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Explanation:
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CHECK POINT!Apopulation has µ = 60 and σ = 5. The distribution of sample means for samples of size n = 4
selected from this population would have an expected value of 60.
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6.6 Finding Probabilities for Sample Means
Remember that the purpose of the distribution of sample means is to find the probability associated with any specific
sample. Previously we learned how to find the probability of obtaining specific scores given a particular distribution
of scores. (eg. IQ scores form a normal distribution with µ = 100 and σ = 15. What is the probability of randomly
selecting and individual with an IQ score greater than 120?) Now, instead of dealing with scores and distributions of
scores, we are dealing with sample means and distributions of sample means. The logic is the same in both cases.
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Example: The population of scores on the CognitiveAbility test forms a normal distribution with µ = 250 and σ =
25. If you take a random sample of n = 25 participants, what is the probability that the sample mean will be greater
than M = 260? So, out of all possible samples means, what proportion have values greater than 260?
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We know the distribution of sample means has a mean of 250 (because µ = 250)
• For n = 25, the distribution has a standard error of σM = σ/√n =25/√25 = 5
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If µ = 250 and σM = 5, what is the corresponding z-score for M = 260?
– In this case, it’s easy to see that z = 2.00
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We can also use formula:
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Note changes from previous z-Score Formula:
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Because this distribution os sample means is normal, we can use the unit normal tableto find the probability
associated with z = 2.00.
1. P (z > 2.00) = 0.0228
2. Therefore, we would conclude that it is very unlikely to obtain a random sample of 25 participants with an
average CognitiveABility Test score greater than 260
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Practice: z-Scores for Sample Means:Asample of n = 25 scores has a mean of M = 84. Find the z-score for this
sample:
1. If it was obtained from a population with µ = 80 and σ = 10.
a. = 2
2. If it was obtained from a population with µ = 80 and σ = 20.
a. = 1
3. If it was obtained from a population with µ = 80 and σ = 40.
a. = 0.5
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Standard Error and Error Bars: It is important to note that error bars do not always represent standard error (may
show standard deviation or confidence intervals). Standard Error bars illustrate the mean plus or minus 1 standard
error.
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CHECKPOINT!Arandom sample of n = 16 scores is obtained from a population with mean = 50 and sigma = 16. If
the sample mean is M = 58, what is the z-Score corresponding to the sample mean? 2.
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Inferential Statistics
1. Will compare sample of treated rats to untreated rats in the original population.
2. How much difference is necessary before we can say that the sample rats is noticeable different from the
population?
3. We use the 95% probability. If it falls within 95%, it is possible by chance, so not statically significant. If it is
outside it, we say the treatment works and it is statistically significant.
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Independent Random Sample
1. The probability of being included in the sample is the same for all members of the population of interest
2. Sampling must be done with replacement.
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7. Introduction to Hypothesis Testing
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HOMER: Hypothesize, Operationalize, Measure, Evaluate, Replicate/Revise/Report.
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Hypothesis Test: Statistical procedure that uses data collected from a sample to evaluate a particular
hypothesis about a population.
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Two Opposing Hypotheses:
1.population.hesis 0H ): The independent variable (Treatment) has no effect on the dependent variable for a
2. Alternative Hypothesis 1H ): The independent variable (treatment) does have an effect on the dependent
variable. This is a non-directional hypothesis.
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Example One: Blood cortisol levels during stress are normally distributed with a µ = 20.65 mcg/dl and σ = 3.24.
Sample of 32 participants, have them perform a difficult task in front of an audience, give them all a chocolate bar,
then measure their cortisol levels.
Null Hypothesis: µ = 20.65 same as original population. It is true until proven otherwise.
Alternative Hypothesis:chocolate0.65. This is a non-directional hypothesis. So we are keeping open the
possibility that chocolate may actually increase cortisol levels.
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Setting the Criteria:
1. Our focus is on whether we will reject or fail to reject our Null Hypothesis
2. We never accept the null hypothesis (or the alternative hypothesis)
3. We must set the criteria for how much evidence we need in order to reject the null hypothesis
b. In a typical psychological study, we state that if there is less than a 5% chance that we would select a sample
from the original population with this sample mean, we can assume that our sample comes from a different
population. So our unknown treated population is different from our known population.
4. Must decide which sample means would be consistent with the null hypothesis, and which sample means would
be at odds with the null hypothesis
5. TheALPHAVALUE, or the LEVEL OF SIGNIFICANCE, is the probability value used to define “very unlikely”
eg. With α = 0.05, we separate the most unlikely 5% of the sample means (extreme values) from the most likely
▯ 95% of the sample means (central values)
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Critical Regions for α = 0.05 and a Non-Directional (or 2-tailed) Hypothesis Test
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The CRITICAL REGION is defined by the alpha level eg.Alpha of 0.05 indicates that the size of the critical region
is p = 0.05
1. It contains those sample values that would be very unlikely to be obtained if our null hypothesis is true
2. Therefore, if our sample data fall within this critical region, we will reject our null hypothesis.
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Note: We always assume that the standard deviation of the population hasn’t changed. We assume that any treatment
effect is going to change the mean (push it up or down) but not the variability of the shape of the distribution.
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Collect Data & Compute Sample Statistics
Back to example one: M = 19.20, σ = 3.11. If the null hypothesis is true, what is the likelihood of us obtaining a
sample mean of M = 19.20? Does this value fall within the critical region?
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Make a Decision:
1.We either decide to reject or fail to reject the null
hypothesis
2.In this case, because our obtained z-statistic falls
within the critical region (or rejection region) re
reject the null hypothesis. (x < -1.96, 1.96 < x)
3.We have enough evidence to conclude that the
treatment does have an effect.
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Statistical Significance
Aresult is said to be statistically significant if it is very unlikely to occur by chance when the null hypothesis is true.
1. If we reject the null hypothesis, we are saying that the treatment had a significant effect.
2. Note that this does not mean that it is a large effect, it just means that it is unlikely.
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Assumptions for Hypothesis Tests with z-Scores
1. The population standard deviation is unchanged by the treatment
2. The sample is randomly selected from the population of interest eg. Not a sample of the most relaxed looking
people I can find
3. The observed are independent eg. The cortisol level for participantAis independent from the cortisol level for
participant B
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Example Two: PrenatalAlcohol and Birth Weight
Aresearcher would like to examine the effect of prenatal alcohol on birth weight. It is known that weight of regular
newborn rats forms a normal distribution with µ = 18 grams and σ = 4. You take a random sample of 16 pregnant
rats and give them a daily dose of alcohol. You then select a random pup from each litter (n = 16) and weigh them.
The average weight for the sample is M = 15 grams. Does alcohol have a significant effect on the weight of newborn
rats? Use a two-tailed test with α = 0.05.
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H 0 µalcohol exposure The weight of the rats exposed to alcohol is no different from the population without
exposure to alcohol.
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H 1 µalcohol exposure The weight of the rats exposed to alcohol is significantly different from the population
without exposure to alcohol
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According to H 0he distribution of sample means is centered at µ = 18, and has a standard erroM of σ = 4/√16. = 1
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Therefore, the treatment with alcohol had a significant effect on the birth weight of new born rats, z = 3.00, p < 0.05.
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Directional (one-tailed) Hypothesis Tests
1. In most cases, a researcher has a directional hypothesis -> a prediction about the direction of the effect ie.
Whether it will increase or decrease values. Eg. That chocolate will decrease cortisol values.
2. In these cases, the researcher may decide to conduct a one-tailed test.
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▯47 /18 University of Toronto - Revision Paper - PSY201 Marcus Lam
Null Hypothesis (0 ) chocolate0.65
Alternative Hypothesis1(H )chocolate0.65
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Critical Region Chart
Consider: What would our critical z-values be for
the one tailed and two-tailed hypothesis tests if α
= 0.01. Higher magnitude, smaller area.
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Can our decision be wrong? YES! Our sample provides limited information about the whole population. There is
always a chance that the sample is misleading, and will cause a researcher to make the wrong decision. Two different
types of errors:
1. Incorrectly rejecting the null hypothesis
2. Incorrectly failing to reject the null hypothesis
Type I Error: Occurs when the researcher rejects a null hypothesis that is actually true.
1. The researcher concludes that a treatment does have an effect when it in fact does not. *Very bad thing! Imagine
a scientist concluding that a treatment has an effect, when really the drug is ineffective
2. How can we reduce changes of Type I error? With the selection of alpha level -> the alpha level for a hypothesis
test is the probably that the test will lead to a Type 1 error.
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Type II Error: Occurs when the researcher fails to reject a null hypothesis that is actually false.
1.committing a Type I error are generally less serious than committing a Type I errort does. *consequences of
2. Probability of committing a Type II error is represented as ß
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Flaw in the Logic of Hypothesis Testing
1. Probabilities eg. 5% chance of Type I error are well defined only when the null hypothesis is true.
2. If the null hypothesis is false, then other factors come into plat (Eg. How large is the treatment effect) and it
becomes impossible to assign precise probabilities
▯. Some argue that the rate of Type I errors in published research is higher than the alpha levels used would suggest.
Practice: Aresearcher is investigating the effectiveness of a new study-skills training program for elementary school
children.Asample of 25 third grade children is selected to participate in the program and each child is given a
standardized achievement test at the end of the year. For the rgular population of third grade chilren, scores on the
test for ma normal distribution with a mean of µ = 150 and σ = 25. The mean for the sample is M = 158. Does the
program have an significant effect on the test scores of the participants? Use a two tailed test with α = 0.05.
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▯48 /18 University of Toronto - Revision Paper - PSY201 Marcus Lam
Effect Size:
1. As the result of a hypothesis test we may conclude that an independent variable has a significant effect on an
dependent variable.
2. But.. This doesn’t tell us how much or an effect.
3. An hypothesis test evaluates the relative size of a treatment effect -> if standard error is very small, the effect of
treatment can be very small, yet still large enough to be significant.
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4. Ameasure of effect size is intended to provide a measurement of the absolute magnitude of a treatment effect,
independent of the size of the sample being used.
5. Cohen’s d: standardizes the effect size by measuring the mean difference in terms of the standard deviation.
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Example: Normal population with µ = 50, σ = 10.
Sample of n = 1000, M = 50.7.
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σ = 10/√1000 = 0.316
z= (50.7-50)/0.316 = 2.215
d = (50.7-50)/10 = 0.07
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15 -point difference in two situations
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