standard deviation is a standardized measure of dispersion, indicating that it can be used when working with a specific type of distribution called the normal curve. The normal curve will be discussed at length in later chapters.
Not all measures of linear dispersion are standardized. One unstandardized measure of dispersion is the mean absolute deviation, which uses the absolute value of each deviation score rather than the squaring technique. Although this provides a more intuitive measure of dispersion, the measure does not fall within known locations on the normal curve, which limits its use.
A statistic (or parameter) that is used to measure dispersion in describing a sample is called a descriptive statistic. Alternatively, you can use sample statistics to make guesses about larger populations. This is commonly referred to as inferential statistics. You are using the sample data to make inferences (guesses) about the larger population.
When using sample variances and standard deviations to infer about a larger population, it is important to note that your estimate will not be precise. In other words, your sample variance likely will not equal the population variance. In fact, you can be almost certain that the sample variance will be less than the population variance. Placing n − 1 in the denominator of the variance and standard deviation formulas will adjust for this bias when estimating the population standard deviation from a sample. This adjustment will increase the variance (or standard deviation), which will help to better approximate the population variance (or standard deviation). Because of the inferential use of the standard deviation and variance that will be introduced later, some instructors prefer to use n – 1 in the denominator of even the descriptive standard deviation and variance formulas. Ask your instructor which formula is preferred.
Computational Exercises
The following are the test grades for students in your European History class after the first test.
1 Find the deviation score for each test grade. What is the sum of these deviations?
2 Find the variance for the grades. Find the standard deviation for the grades.
3 How many standard deviations from the mean is the person with the highest grade? The person with the lowest score?
4 One of those who received a 99 is an exceptional student who turned in an extra-credit project that was worth an additional 10 points on the test. What would the variance and standard deviation be with this person’s correct grade? (Recalculate the measures of dispersion using this new high score.)
5 Reflecting back on your response to Question 4, which measure of dispersion had the largest change in absolute units?
6 How many of the original grades are between 1 and 2 SD above the mean?
The manager of a shoe store is interested in determining how many of each shoe size were sold the previous day. The store has made 10 sales of shoes with the following sizes:
7. Find the variance and standard deviation for these shoe sizes.
8. In the last few minutes of the store’s business hours, three people run in, stating that they are in a shoe emergency, and ask to purchase shoes. The shoe sizes of these three new customers are 8, 6, and 10. If they each purchase a pair of shoes, what will be the new standard deviation?
9. How many of these 13 (including those added in Question 8) people fall within 1 SD of the mean?
10. What is the mean absolute deviation of the original shoe sizes? How does this compare with the standard deviation?
11. If you were to estimate the population variance in the shoe sizes of people who shop at this store from this sample, what formula would you use (using the original 10 scores)? What would be the population variance estimated from this sample?
12. The store manager collects similar data for the following day and finds that the mean is substantially higher but the standard deviation is now 7.9. Which mean is more representative of the average shoe size of those who shop at this store?
Computational Answers
1
2 The sum of the deviations is 0.00. Variance = 126.99 (using N) or 133.67 (using n − 1); standard deviation = 11.27 (using N) or 11.56 (using n − 1).
3 The highest grade is approximately 1.20 standard deviations above the mean. The lowest grade is approximately 1.7 SD below the mean.
4 The new variance = 145.64 (using N) or 153.31 (using n − 1). The new standard deviation = 12.07 (using N) or 123.38 (using n − 1).
5 The variance was the most affected by this change.
6 Three scores: 98, 99, and 99.
7 The variance = 6.60 (using N) or 7.33 (using n − 1). The standard deviation = 2.57 (using N) or 2.71 (using n − 1).
8 The new standard deviation would be 2.42 (using N) or 2.52 (using n − 1).
9 There are seven scores that fall within 1 SD above and below the mean.
10 You would use the formula with n − 1 in the denominator because the sample size is less than 30. The estimated population variance would be 7.33.
11 The first mean is more representative because the scores are more tightly clustered about the mean.
True/False Questions
1 The range is a very sensitive measure of central tendency.
2 Adding 25 different scores to the center of a data set will affect the range.
3 A deviation score provides a measure of the score’s distance from the mean.
4 The sign (+ or −) of a deviation score indicates its location in relationship to the mean.
5 Deviation scores always sum to 1.
6 The variance is the square of the average distance of scores from the mean, measured in squared distance units.
7 Standard deviations from different samples of the same population can be compared.
8 The standard deviation is calculated from the square root of the variance to revert the measure to linear units.
9 The standard deviation is the most commonly used measure of dispersion because of its interpretability and applicability to the normal curve.
10 The mean absolute deviation is used more frequently than the standard deviation.
11 Descriptive statistics are used to describe the characteristics of a sample.
12 The mean and standard deviation are examples of inferential statistics.
13 When using a sample to infer about a population, you should use n − 1 in the denominator.
