Variability
A quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together
The two purposes of variability
1.) Variability describes the distribution: specifically, it tells whether the scores are clustered close together or are spread out over a large distance. Usually, variability is defined in terms of distance. It tells how much distance to expect between
What are the 3 measures of variability?
1.) Range
2.) Standard Deviation
3.) Variance
(Of the 3, standard deviation and variance are the most important)
Range
Distance covered by the scores in a distribution, from the smallest score to the largest score.
The most commonly used definition of the range simply measures the difference between the largest score (XMax) and the smallest score (XMin). Many computer pro
Problem with using Range as a measure of Variability
It is completely determined by the two extreme values and ignores the other scores in the distribution
Standard Deviation
Most commonly used and important measure of variability
Standard deviation uses the mean of the distribution as a reference point and measures variability by considering the distance between each score and the mean
In simple terms, the standard deviation
Example of How to find the Variance and Standard Deviation from a population
N = 5 scores
1, 9, 5, 8, 7
Step 1.)
Compute the mean. The five scores add up to 30.
30/5 = 6 = The Mean
Step 2.)
Find the deviation of each score
X - The Mean
1 - 6 = -5
9 - 6 = 3
5 - 6 = -1
8 - 6 = 2
7 - 6 = 1
Next, square each value to get 25, 9, 1, 4,
Standard Deviance and Variance can only be used for ____________ or ____________ scales
Interval; Ratio
This is because standard deviance and variance measure distance so in order for it to work there must be an equal distance between each score.
Finding the Sum of Squared Deviation Scores (SS) using the Definitional Formula
Definition Formula of SS = ?(X-�)�
To find the sum of the squared deviations, the formula instructs you to perform the following sequence of calculations
Step 1.) Find each deviation score (X - the Mean)
Step 2.) Square each deviation score
Step 3.) Add u
Finding the Sum of Squared Deviation Scores (SS) using the Computational Formula
When the mean is not a whole number, the deviations all contain decimals or fractions, and the calculations become difficult.
In addition, calculations with decimal values introduce the opportunity for rounding error, which can make the result less accura
Population Variance Symbol
?�
Population Standard Deviation Symbol
?
Sample Variance Symbol
s�
Sample Standard Deviation Symbol
s
True or False:
Calculating SS for a sample is exactly the same as for a population, except for the minor changes in symbols used
True
Sample Variance
Obtained by dividing the sum of squares by n-1
Sample Variance = SS/(n-1)
Sample Standard Deviation Formula
s = ?SS/(n-1)
Sample variance is often called _____________ _______________ _____________
Estimated population variance
Sample standard deviation is often called _____________________
Estimated population standard deviation
Degrees of Freedom
For a sample of n scores, the degrees of freedom, or df, for the sample variance are defined as df = (n-1).
The degrees of freedom determine the number of scores in the sample that are independent and free to vary
If sample variance is computed by dividing by n, instead of n ? 1, how will the obtained values be related to the corresponding population variance?
They will constantly underestimate the population variance
By dividing by n-1 instead of dividing by n we get a sample variance that provides a much more accurate representation of the population variance.
Unbiased estimate
A sample statistic is unbiased if the average value of the statistic is equal to the population parameter. (The average value of the statistic is obtained from all the possible samples for a specific sample size, n.)
This does not mean that each individua
Biased estimate
A sample statistic is biased if the average value of the statistic either underestimates or overestimates the corresponding population parameter.
Which of the following is an example of an unbiased statistic.
both the sample mean and the sample variance (dividing by n ? 1)
True or False:
Adding a constant to each score changes both the mean and the standard deviation
False, it only changes the mean
True or False:
Multiplying each score by a constant changes both the mean and the standard deviation
True
Research reports typically summarize the data by reporting only the __________ and the standard deviation
Mean
According to the book, an extreme score is any score more than ______ standard deviations away
from the mean
2
As a general rule, about _____ of the scores will be within one standard deviation of the
mean, and about _____ of the scores will be within two standard deviations of the mean
70%; 95%
z-score
A measure of how many standard deviations a score is away from the mean
It's primary purpose is to describe the exact location of a score within a distribution.
A z-score specifies the precise location of each X value within a distribution. The sign of th
Z-Score Formula
z = (x - ?)/?
a.k.a.
z = the deviation score divided by the standard deviation
Formula to find the raw score from the Z-Score
X = ? + z?
True or False:
The distribution of z-scores will have exactly the same shape as the original distribution of scores.
True
The z-score distribution will always have a mean of _________-
0
The distribution of z-scores will always have a standard deviation of ____________
1
Standardized Distribution
Composed of scores that have been transformed to create predetermined values for the mean and the standard deviation.
Standardized distributions are used to make dissimilar distributions comparable
What is the process for standardizing a distribution to create new values for the mean and the standard deviation?
Step 1.) Transform the original raw scores into z-scores
Step 2.) Transform the z scores into new X values so that the specific mean and standard deviation are attained