Class Notes
(806,513)

Canada
(492,267)

University of Alberta
(12,903)

Statistics
(237)

STAT151
(146)

Susan Kamp
(11)

Lecture

# 6.pdf

Unlock Document

University of Alberta

Statistics

STAT151

Susan Kamp

Fall

Description

Ch 6 The Standard Deviation as a Ruler
and the Normal Model
Shifting data:
- Adding (or subtracting) a constant to every data value adds
(or subtracts) the same constant to measures of position.
- Adding (or subtracting) a constant to each value will
increase (or decrease) measures of position: center,
percentiles, max or min by the same constant.
- Its shape and spread - range, IQR, standard deviation -
remain unchanged.
Example: You have a data set: y = 1, y =12, y = 2, y = 4,3y = 4 5
5. If you want to add a constant c = 2 to each observation in this
data set, how does it affect the mean, median, Q , Q , ma1, mi3,
range, standard deviation, IQR, and its shape?
Summary statistics:
Column n Mean Variance Std. Dev. Std. Err. Median Range Min Max Q1 Q3
y 5 3 2.5 1.5811388 0.70710677 3 4 1 5 2 4
y + 2 5 5 2.5 1.5811388 0.70710677 5 4 3 7 4 6
1 of 21 Summary:
If y = y + c for each observation
new original
- For measures of center or position:
Center Center c
o new original
o Position new Position original
- For measures of spread and shape:
o Spread new Spread original
Shape Shape
o new original
Rescaling Data:
- When we multiply (or divide) all the data values by any
constant, all measures of position (such as the mean,
median, and percentiles) and measures of spread (such as
the range, the IQR, and the standard deviation) are
multiplied (or divided) by that same constant.
Example: You have a data set: y =11, y =22, y =33, y =44, y =5
5. If you want to multiply each observation with a constant d =
2 of 21 2 in this data set, how does it affect the mean, median, Q , Q1, 3
max, min, range, standard deviation, and IQR?
Summary statistics:
Column n Mean Variance Std. Dev. Std. Err.Median Range Min Max Q1 Q3
y 5 3 2.5 1.5811388 0.70710677 3 4 1 5 2 4
y * 2 5 6 10 3.1622777 1.4142135 6 8 2 10 4 8
Summary:
y d y
If new originalfor each observation
- For measures of center or position:
Center d Center
o new original
Position d Position
o new original
- For measures of spread and shape:
Spread d Spread
o new original
Shape Shape
o new original
3 of 21 Thus, if you rescale and shift data:new d y original for
each observation
- For measures of center or position:
o Center new d Center originalc
Position d Position c
o new original
- For measures of spread and shape:
o Spread new d Spread original
o Shape new Shape original
Example: Students taking an intro stats class reported the
number of credit hours that they were taking that quarter.
Summary statistics are shown in the table.
Mean Std Dev Min Q1 Median Q3 Max
16.65 2.96 5 15 16 19 28
Suppose the college charges $73 per credit hour plus a flat fee of
$35 per quarter. For example, a student taking 12 credit hours
would pay $35 + 12($73) = $911 for that quarter.
What is the mean fee paid?
4 of 21 What is the standard deviation for the fees paid?
What is the median fee paid?
Use of Standard Deviation
The distance to the mean of a specific observation measured in
standard deviations gives information about the location of this
observation in relation to the other observations in the sample.
If you obtained a score in an achievement test you might want to
know your standing in relation to other people who have taken
the test. Are you below or above the mean, how far above or
below in relation to the other people.
Standardizing with z-score
z-score or standardized value is a measure of relative standing.
If x is an observation from a sample with mean y and standard
deviation s, then the standardized value or z-score of y is
y y
z s
5 of 21 - tells “how many standard deviations away from the mean
does the measurement lie and in which direction?”
o The standardization makes numbers from different
contexts comparable.
o Positive z-score
o Negative z-score
o z-score of 0
NOTE: standardized values have no units.
Benefits of Standardizing
Standardized values have been converted from their
original units to the standard statistical unit of standard
deviations from the mean.
Thus, we can compare values that are measured on
different scales, with different units, or from different
populations.
Standardizing data into z-scores shifts the data by
subtracting the mean and rescales the values by dividing by
their standard deviation.
Standardizing into z-scores does not change the shape
of the distribution.
6 of 21 Standardizing into z-scores changes the center by
making the mean 0.
Standardizing into z-scores changes the spread by
making the standard deviation 1.
Example:
In a class of 3 students, the average score on the final exam is
75% and standard deviation of 5%. Amy got 80% on the final
exam. How many standard deviations better than the mean is
that?
Mandy got 70%. How many standard deviations is Mandy’s
score deviate from the mean?
Judy got 75%. How many standard deviation is Judy’s score
deviate from the mean?
7 of 21 NOTE:
1) standardizing does not change the shape of the distribution
of a variable
2) standardizing changes the center by make the mean 0
3) standardizing changes the spread by making the standard
deviation 1.
Example:
Two graduate students:
- accounting major gets job offer for $ 35000
- advertising major gets job offer for $ 33000
other students of:
- accounting: y = 36000 and s = 1500
- advertising: y = 32500 and s = 1000
Who do you think is happier about his/her job offer?
8 of 21 Normal Model
Many numerical variables have bell shaped histograms. For
example, heights, weights, lifetime of a light bulb, etc. The
normal distribution provides a reasonable approximation for
modeling this type of data. It is the most important and most
widely used of all probability distributions.
Properties of normal distributions.
- These curves are symmetric, unimodal, and bell-shaped.
- For every combination of a mean and a standard
deviation , there is a different curve.
o is the center of the distribution (right at the highest
point of the density distribution function)
o controls the spread of the distribution.
o Notation: N(μ,σ) represents a Normal model with a
mean of μ and a standard deviation of σ.
9 of 21 Recall:
1)Population parameter is a numerical measure such as the
mean, median, mode, range, variance, or standard deviation
calculated for a populat

More
Less
Related notes for STAT151