Lesson 1.3 · Unit 1 · Science practice skills 3.A and 3.B

Describing data: averages, spread, and the normal curve

Psychologists collect numbers, and a pile of numbers means nothing until it is summarized. Descriptive statistics do that job: they describe the data you have, using a measure of where the scores are centered, a measure of how spread out they are, and a picture of their overall shape. The exam asks you to calculate the simple ones from a small data set and to interpret all of them, including the standard deviation and percentile rank, which you will not have to compute from raw data but must be able to read. This lesson gives you everything you need for that, with careful practice, because these items are among the most reliable points on the test.

Required skills · 3.A, 3.B

Science practice skills.

3.A. Identify psychology-related concepts in descriptions or representations of data.
3.B. Calculate and interpret measures of central tendency, variation, and percentile rank in a given data set.

What the exam asks you to do.

3.A: Identify variables in descriptions or representations of data.
3.A: Identify statistical and psychological concept(s) depicted in a table, graph, chart, or figure.
3.B: Calculate mean, median, mode, and range from a set of data.
3.B: Explain the elements of the normal curve including percentages of distributions of scores across the curve, how to interpret skewness in a graph, and how to interpret a bimodal distribution.
3.B: Interpret mean, median, mode, range, standard deviation, and percentile rank from a set of data.
3.B: Explain how regression toward the mean occurs as more data are collected.

Measures of central tendency

A measure of central tendency gives one number that represents the center of a distribution.

The mean is the arithmetic average: add all the scores and divide by the number of scores.
The median is the middle score when the scores are put in order. If there is an even number of scores, it is the mean of the two middle scores. Half the scores are above it and half are below.
The mode is the most frequent score. A distribution can have no mode, one mode, or more than one.

The three can differ, and which is best depends on the data. The mean uses every score, so it is pulled strongly by extreme scores called outliers. The median is not affected much by outliers, so it is the better summary when a few extreme values would distort the mean, which is why reports of household income use the median.

Measures of variation

Two groups can have the same mean and look nothing alike, so a measure of variation says how spread out the scores are.

The range is the highest score minus the lowest. It is easy to compute but depends only on the two most extreme scores.
The standard deviation (SD) describes the typical distance of scores from the mean. A small SD means the scores cluster tightly around the mean; a large SD means they are widely scattered. You will be asked to interpret an SD, not to compute one: if two classes both average 75 on a test but class A has an SD of 4 and class B has an SD of 15, class A's scores are all close to 75, while class B has many scores far above and far below.

Shapes of distributions

A graph of a distribution (most often a histogram, which shows how many scores fall in each interval) has a shape, and the shape tells you how the mean, median, and mode relate.

Figure 1.1. Three shapes. In a positively skewed distribution the long tail points toward high scores, and the mean is pulled toward the tail, above the median. A normal curve is symmetric. A bimodal distribution has two peaks, which often means two different groups are mixed together.

In a normal distribution the scores are symmetric around the center and form a bell-shaped curve; the mean, median, and mode are all the same. In a skewed distribution one tail is long. The skew is named for the direction of the tail: a positively skewed distribution has its long tail toward high scores (a few high outliers), and a negatively skewed distribution has its tail toward low scores. In either case the mean is pulled toward the tail, so in a positive skew the mean is greater than the median, and in a negative skew the mean is less than the median. A bimodal distribution has two peaks. It suggests that two different groups are mixed in the data, such as scores from two very different kinds of students, and the mean in that case may describe almost nobody.

The normal curve

Many psychological measures (heights, many test scores, IQ) form a distribution close to the normal curve. The curve has a fixed relationship between the mean, the SD, and the percentage of scores, shown in Figure 1.2.

Figure 1.2. The normal curve. About 34% of scores fall between the mean and 1 SD on each side, 68% within 1 SD, 95% within 2 SD, and nearly all within 3 SD. The percentile rank under each mark is the approximate percent of scores at or below that value.

Memorize the 68-95-99.7 rule: about 68% of scores are within 1 SD of the mean, about 95% are within 2 SD, and about 99.7% are within 3 SD. Because the curve is symmetric, half of those percentages are on each side: about 34% of scores lie between the mean and 1 SD above, 34% between the mean and 1 SD below, and about 13.5% in each band from 1 to 2 SD away.

Percentile rank

A percentile rank tells where a score stands compared with the others: the percent of scores in the distribution at or below that score. A score at the 84th percentile is higher than or equal to 84% of the scores. On a normal curve, the mean is at the 50th percentile, a score 1 SD above the mean is at about the 84th percentile (50% below the mean plus 34% between the mean and 1 SD), and a score 2 SD above is at about the 98th (50% + 47.5%). A score 1 SD below the mean is at about the 16th percentile.

Regression toward the mean

Suppose a coach picks the 10 students who scored lowest on a first quiz and gives them a tutoring session, and on the second quiz their scores rise. Did the tutoring work? Maybe, but there is another explanation. Any score is partly skill and partly luck (a good or bad day, a lucky guess, a missed question). People who score at an extreme the first time are likely to have had unusual luck, and on the next try their luck is likely to be more ordinary. So extreme scores tend to move closer to the average when measured again. This is regression toward the mean, and it shows up as more data are collected. It explains why a very tall couple's children are, on average, a bit shorter than the parents, why a rookie with a spectacular first season often has a weaker second, and why a "worst-ever" week at work is usually followed by a better one. When you evaluate a claim of improvement that follows an extreme score, ask whether a comparison group with no treatment improved as well. In an experiment with a control group, regression toward the mean affects both groups equally.

Worked example · Mean, median, mode, and range

Nine students scored 62, 70, 70, 74, 78, 81, 85, 90, and 98 on a unit test. Find the four statistics.

Step 1, the scores are already in order and there are 9 of them.

Step 2, the mean: add the scores. $62+70+70+74+78+81+85+90+98=708$. Divide by 9: $708\div 9\approx 78.7$.

Step 3, the median: with 9 scores the middle score is the 5th, which is 78.

Step 4, the mode: 70 appears twice and no other score repeats, so the mode is 70.

Step 5, the range: $98-62=36$.

Step 6, interpret: the mean (78.7) is slightly above the median (78), so the distribution is almost symmetric with a very slight pull toward the high scores.

Worked example · How an outlier changes the measures

Eight students reported the minutes they spent on homework one night: 24, 25, 27, 27, 28, 29, 30, and 95. Find the mean and median with and without the 95. Which better represents a typical student?

Step 1, with all eight scores: the sum is 285, so the mean is $285\div 8\approx 35.6$ minutes. The median is the mean of the 4th and 5th scores, $(27+28)\div 2=27.5$ minutes. The mode is 27.

Step 2, without the 95 (seven scores): the sum is 190, so the mean is $190\div 7\approx 27.1$, and the median is the 4th score, 27.

Step 3, compare: one outlier moved the mean by about 8.5 minutes, but moved the median by only half a minute.

Step 4, conclusion: the median (27.5) describes a typical student far better than the mean (35.6), because seven of the eight students spent between 24 and 30 minutes. The distribution is positively skewed, and the mean (35.6) is greater than the median (27.5), as expected.

Worked example · Percentile rank on a normal curve

A test is scored so that the mean is 100 and the SD is 15, and the scores are normally distributed. Estimate the percentile rank of scores of 115, 130, and 85, and the percent of people who score between 85 and 115.

Step 1, 115 is 1 SD above the mean. Everything below the mean is 50% of scores, and the band from the mean to +1 SD adds about 34%, so the percentile rank is about 84.

Step 2, 130 is 2 SD above the mean: $50+34+13.5=97.5$, so about the 98th percentile.

Step 3, 85 is 1 SD below the mean: $50-34=16$, so about the 16th percentile.

Step 4, between 85 and 115 is within 1 SD of the mean, so about 68% of people.

Step 5, check by symmetry: the 84th percentile (115) and the 16th percentile (85) are the same distance from 50 in opposite directions, and $84-16=68$.

Exam focus

Calculate mean, median, mode, and range from a short list; put the scores in order first.
Outliers pull the mean, so with skew the median is the better center. Mean greater than median means positive skew; less than means negative skew.
A larger standard deviation means more spread. Interpret it; do not compute it.
Use the 68-95-99.7 rule and symmetry to find percentages on a normal curve; a score 1 SD above the mean is at about the 84th percentile.
Percentile rank is the percent of scores at or below a score. Regression toward the mean: extreme scores tend to be less extreme on retest.

Practice

Find the mean, median, mode, and range of 4, 6, 6, 7, 9, 12, and 14.
Show answer
The sum is 58 and there are 7 scores, so the mean is $58\div 7\approx 8.3$. The median is the 4th score, 7. The mode is 6. The range is $14-4=10$.
A distribution of salaries at a small company has a mean of $82,000 and a median of $58,000. Describe its shape and say which measure better represents a typical salary.
Show answer
The mean is much higher than the median, so the distribution is positively skewed (a few very high salaries pull the mean up). The median, $58,000, better represents a typical salary.
Two sections of AP Psychology both have a mean of 80. Section A has an SD of 3 and section B has an SD of 12. Which section's scores are more spread out, and what does that mean for the students?
Show answer
Section B has the larger SD, so its scores are more spread out: many students scored far above and far below 80. Section A's students scored very close to 80.
On a normal curve with a mean of 50 and an SD of 10, about what percent of scores are between 40 and 60? Between 50 and 70?
Show answer
40 to 60 is within 1 SD of the mean, so about 68%. 50 to 70 is the mean to +2 SD, which is about $34+13.5=47.5\%$.
A score is at the 84th percentile. What does that mean?
Show answer
About 84% of the scores in the distribution are at or below it (it is higher than or equal to most of the group).
A new reading program is given to the 15 lowest-scoring students in a class, and their average score rises by 6 points on the next test. Give one reason the gain might not come from the program.
Show answer
Regression toward the mean: extreme low scores on the first test partly reflect bad luck, so on a retest these students would be expected to score closer to the average even with no program. A control group of equally low scorers who did not get the program would show how much of the gain is due to this effect.