Statistics Midterm 2 Review
CHAPTER 6: The Normal Curve, Standardization and Z scores
Normal Curve: a specific bell-shaped curve that is unimodal, symmetric and
Standardization, z Scores, and the Normal Curve
Standardization: Converts individual scores from different normal
distributions to a shared normal distribution with a known mean, standard
deviation, and percentiles.
The Need for Standardization
Z Score: the number of standard deviations a particular score is from the
Transforming Raw Scores into Z Scores.
(X - m)
Transforming Z scored into Raw Scores
X = z(s)+m
Z distribution: a normal distribution of standardized scores.
Standard normal distribution: a normal distribution of z scores.
The standardization distribution allows us to do the following:
1. Transform raw scores into standardized scores called z scores.
2. Transform z scored back into raw scores
3. Compare z scores to each other—even when the z scores represent
raw scores on different scales.
4. Transform z scored into percentiles that are more easily understood.
Transforming z Scores into Percentiles.
Z scores are useful because:
1. They give us a sense of where a score falls in relation to the mean of
its population (in terms of the standard deviation of its population)
2. Z scores allow us to compare scores from different distributions
3. Z scores can be transformed into percentiles.
1 Statistics Midterm 2 Review
The Central Limit Theorem
Central limit Theorem: refers to how a distribution of sample means is a
more normal distribution than a distribution of scores, even when the
population distribution is not normal.
The central limit theorem demonstrates two important principles:
1. Repeated sampling of means approximated a normal curve, even
when the original population is not normally distributed.
2. A distribution of mean is less variable than a distribution of
Distribution of means: a distribution composed of many means that are
calculated from all possible samples of a given size, all taken from the same
The mean of the sample means will be the same as the mean.
Standard error: the name for the standard deviation of a distribution of
The standard error measures (roughly) the average difference between M
and μ that should occur by random sampling alone (i.e., roughly, the average
value for M - µ).
Formula for standard error:
s M =
You can find sample means from z-scores using:
M = µ + (z)(σ )
Chapter 7: Hypothesis Testing with z Tests
Hypothesis Testing with Z Tests
Z test: a hypothesis test in which we compare data from one sample to a
population for which we know the mean and the standard deviation.
Given a score, Find a proportion
a) Finding Proportions and Probabilities
Step 1 – Sketch the normal distribution and shade the target area
Step 2 – Choose the appropriate z-table column
Step 3 – Convert X (raw score) to a Z-score
Step 4 – Use the z-table to find the proportion (probability) of your raw score (X)
2 Statistics Midterm 2 Review
Raw scores, z Scores, and Percentages
The z table allows us to translate the standardized z distribution into
percentages and individual z scores into percentile ranks.
We can determine the percentage associated with a given z statistic by
following two steps.
o Step 1: Convert raw score into a z score.
o Step 2: Look up a given z score on the z table to find the percentage of
scores between the mean and that z score.
Steps to finding percentages:
o Step 1: Convert the raw score to a z score
o Step 2: Look up the z score on the z table to find the associated
percentage between the mean and the z score.
o Once we know that the associated percentage is 33.65%, we can
determine a number of percentages related to the z score.
Calculating the percentile for a positive z score: We add
50% to the percentage between the mean and that z score to
get the total percentage below that z score.
Calculating the percentage above a positive z score: We
subtract the percentage between the mean and that z score
from 50% to get the percentage above that z score.
Calculating the percentage at least as extreme as our z
score: For a positive z score, we double the percentage above
that z score to get the percentage of scores that are at least as
Calculating a score from a percentile: We can convert a
percentile to a raw score by calculating the percentage
between the mean and the z score, and looking up that
percentage on the z table to find the associated z score. We
would then convert the z score to a raw score using the formu
The Assumptions of Hypothesis Testing
Assumption: A characteristic that we ideally require the population from
which we are sampling to have so that we can make accurate inferences.
Parametric Test: an inferential statistical analysis based on a set of
assumptions about the population.
Nonparametric test: an inferential statistical analysis that is not based on a
set of assumptions about the population.