Understanding Statistical Variability and Distributions
Understanding Measures of Variability
Measures of variability indicate how much scores in a distribution vary, either from the mean or across the full extent of the distribution. It represents the spread of all the scores. Four measures of variability are discussed here: the range, the average mean deviation, the variance, and the standard deviation. These measures can reveal the consistency or similarity of the scores in a distribution and the extent to which the mean truly represents all of the scores.
The Range
The range is a measure of the full extent to which the scores are spread out in a distribution, from the highest to the lowest. It is the easiest measure of variability to compute, but it is seldom used because of its instability. One extreme score can drastically alter the range.
Range = High Score – Low Score
Average Mean Deviation (AMD)
The Average Mean Deviation (AMD) is the average deviation of each score from the mean of the distribution. To compute the average mean deviation, the mean is subtracted from each score to arrive at the deviation scores. The deviation scores are summed and then divided by the total number of scores. Since about half of the deviation scores are positive (for scores larger than the mean) and the other half are negative (for scores smaller than the mean), the average mean deviation is always equal to zero. Thus, because the result is always zero, it cannot be used for any meaningful comparisons between different distributions.
Variance
Variance is represented by either 
(population) or 
(sample). Variance is computed from deviation scores that are squared to remove the positive and negative signs. The mean of the squared deviation scores is called the variance.
Population variance is the actual computed variance of the population. Sample variance is an estimate of the population variance using only the values in the sample. Because samples rarely contain all the extreme scores in a population, their variances are generally smaller than those of the population. To be a good estimate of the variance of the population, a correction factor of n – 1 is used to increase the variance of a sample.
Standard Deviation
Standard deviation is represented by either 
(population) or S (sample). The standard deviation is the square root of the variance.
It represents the average amount each score in the distribution deviates from the mean. It is expressed in the same units as the original scores and indicates the consistency or similarity of the scores in a distribution and the extent to which the mean truly represents all of the scores.
S = 
Note:
- Deviation scores indicate how far a score is from its mean, and they sum to zero.
- Variance is a step towards calculating the standard deviation.
- The larger the standard deviation, the more spread out the results are.
- The smaller the standard deviation, the more clustered the results are.
Scaled Scores and Standard Scores (Chapter 6 Concepts)
Review the following key terms.
Scaled Scores
Scaled scores are scores that are adjusted via some type of scale by applying the same constant to all the scores in the distribution.
Standard Scores (z-scores)
Standard scores or z-scores are uniform values to which any raw score value can be converted. A z-score indicates the number of standard deviation units a raw score is positioned either above or below the mean.
Z-Score Distribution
The z-score distribution is a distribution of transformed raw scores that has a mean of zero and a standard deviation of 1. The transformation of a distribution of raw scores into a distribution of z-scores is done so comparisons between different distributions can be made.
The Normal Curve and Distribution (Chapter 7 Concepts)
Normal Curve Characteristics
A normal curve is the graphic description of the normal distribution. Since all scores are represented under the normal curve, the entire area is equal to 1. Thus, the normal curve can be envisioned as a picture of the proportion of the total number of scores lying under the curve. While normal curves can assume various shapes, they all share the following five common characteristics:
- The curve is bell-shaped and symmetrical.
- The mean, median, and mode are all equal.
- The highest frequency is in the middle of the curve.
- The frequency gradually tapers off as the scores approach the ends of the curve.
- The curve approaches, but never meets, the abscissa at both the high and low ends.
Normal Distribution
The normal distribution is the theoretical distribution of scores with most of the scores clustered around the middle (mean, median, and mode are equal) and the frequency of the scores gradually lessening on either side. When graphed, the normal distribution is referred to as the normal curve.
Percentiles and Quartiles
A percentile is the raw score that has a specific percentage of scores falling below it. The mean is the 50th percentile.
Quartiles are the raw scores that divide the distribution into four equal quarters.
Example Calculation: Finding a Score from a Percentile
Question: What score is the 73rd percentile?
Steps:
- Find the z-score corresponding to the 73rd percentile (area = 0.73). From a z-table, z ≈ 0.6128.
- Use the formula: X = μ + (z * σ) or X = Mean + (z * Standard Deviation)
- Example: If Mean (μ or x̄) = 100 and Standard Deviation (σ or s) = 20:
- X = 100 + (0.6128 * 20) = 100 + 12.256 = 112.256
Note on z-scores and raw scores:
- If z = -1.5, Mean = 100, SD = 10, then X = 100 + (-1.5 * 10) = 85
- If z = 3.5, Mean = 100, SD = 10, then X = 100 + (3.5 * 10) = 135
Chapter Summaries and Key Tasks
Chapter 5: Measures of Variability Summary
This chapter focuses on understanding how numbers in a distribution are spread out, either from the mean or across the entire range.
Key Tasks:
- Understand how to calculate the Range.
- Calculate mean, variance, and standard deviation for a list of numbers.
- Explain what a standard deviation means relative to the mean.
Example: If the mean score on a test is 70 points and the standard deviation is 10, what is the range of scores for students who scored within 1 standard deviation of the mean?
Answer: 60 points to 80 points (Mean ± 1 SD = 70 ± 10).
Chapter 6: Scaling Scores and Standard Scores Summary
This chapter covers scaling scores and calculating z-scores.
Key Tasks:
- Know the rules for scaling scores and what happens to the mean and standard deviation when adding, subtracting, multiplying, or dividing scores by a constant.
- Know how to calculate z-scores.
Example: If you have a mean of 100 and a standard deviation of 10, how do these numbers change if you scale by adding a constant of 2 to each number in the distribution?
Answer: The mean becomes 102, and the standard deviation stays the same (10).
Important: Remember the computational formulas for variance and standard deviation, as well as the formula for z.
Chapter 7: The Normal Curve Summary
Read through this chapter to understand the meaning of the Normal Curve.
Key Tasks:
- Memorize the characteristics of the normal curve.
- Know how to read the z-table.
- Know how to calculate areas and percentages under the curve.
- Know how to read the P-table (Percentile table).
- Know how to calculate percentiles and corresponding raw scores.