The Mean and Standard Deviation

THE MEAN AND STANDARD DEVIATIONProbably the two most useful descriptive statistics in psychological research.

The Mean

Themeanof set of scores (abbreviatedM) is the sum of the scores divided by the number of scores. Along with the median and the mode, the mean is just one measure of thecentral tendencyof a set of scores, but the mean is by far the most common and the most useful.

The Standard DeviationAlthough the mean tells us where the center of a set of scores is, it does not tell us how variable those scores are. For example, the scores 45, 55, 50, 53, and 47 have a mean of 50. But the scores 20, 80, 50, 38, and 62 also have a mean of 50. Note, however, that the second set of scores is much more variable than the first set. So in addition to a measure of central tendency, we need a measure ofvariability.

Thestandard deviation(abbreviatedSD) is, roughly, the average amount by which the scores in a setdifferfrom the mean. The scores in the first set above differ from the mean (50) by 5, 5, 0, 3, and 3. So on average, they differ from the mean by a shade over 3. The scores in the second set differ from the mean (again 50) by 30, 30, 0, 12, and 12. So on average, they differ from the mean by about 17. The second set of scores has a greater standard deviation, which reflects the greater variability among the scores.

Unfortunately, the standard deviation is not just the mean difference between the scores and the mean. It is just a bit more complicated. Here is how to compute it. 1) Find the mean. 2) Subtract the mean from each score (or each score from the mean; it does not matter). 3) Square each of these differences. (Remember that the square of a negative number is positive.) 4) Find the mean of these squared differences. 5) Find the square root of this mean. Voila! You have the standard deviation.

For now, when you compute a standard deviation, do it using a table like the one below. Note thatXrefers to the scores in the set, which appear in the first column. The mean of this set of scores is 6, which appears in the every row of the second column. The difference between each score and the mean appears in the third column. The squares of each of these differences appear in the fourth column. Below the fourth column, I have computed the mean of the squared differences and taken the square root of this mean. (Raising a number to the power is the same as taking its square root.) So the standard deviation of this set of scores is 2.45, meaning that the scores differ from the mean of 6 by an average of about 2.45.

XMXM(XM)2

561 1

462 4

86+2 4

26416

86+2 4

76+1 1

26416

96+3 9

76+1 1

86+2 4

X= 60(X M)2 = 60

M= 60 / 10 = 6SD2 = 60 / 10 = 6

SD= 61/2 = 2.45

The Importance of the Standard DeviationThere are a lot of reasons that it is important to consider the standard deviation of a set of scores. Probably the most important is that it gives us interesting information about the scores that the mean alone does not give. For example, consider an exam in a research methods course. In one section of the course, the mean score was 70 and the standard deviation was 4. This means that overall the students did OK (indicated by the mean of 70) and also that most of them tended to score pretty close to 70 (indicated by the standard deviation of 4). So most of the students did OK. In another section of the course, the mean score was also 70 but the standard deviation was 20. Although one still might say that overall the students did OK (indicated by the mean of 70), the high standard deviation indicates that there was a lot of variability. Some students must have scored quite low and others must have scored quite high. So even though the means were the same, the standard deviations paint very different pictures of student performance on these exams.

Another example: You may have heard that people who work in the field of traffic safety say that what makes a stretch of highway dangerous is not how fast people drive, but how variable their speed is. For example, a stretch of highway on which peoples mean speed is 80 will be fairly accident free if the standard deviation is, say, 3. This is because there is not much variability; people are all clipping along at close to the same speed. But a stretch of highway on which peoples mean speed is 65 might be quite dangerous if the standard deviation is 12. This is because some people must be driving considerably slower than 65 while others must be driving considerably faster. This is likely to lead to lots of braking and lane changing, which makes accidents more likely.

WHAT AP-VALUE TELLS YOU ABOUT STATISTICAL DATAByDeborah J. RumseyfromStatistics For Dummies, 2nd EditionWhen you perform a hypothesis test in statistics, ap-value helps you determine the significance of your results. Hypothesis tests are used to test the validity of a claim that is made about a population. This claim thats on trial, in essence, is called thenull hypothesis.Thealternative hypothesisis the one you would believe if the null hypothesis is concluded to be untrue. The evidence in the trial is your data and the statistics that go along with it. All hypothesis tests ultimately use ap-value to weigh the strength of the evidence (what the data are telling you about the population). Thep-value is a number between 0 and 1 and interpreted in the following way:

A smallp-value (typically 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.

A largep-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

p-values very close to the cutoff (0.05) are considered to be marginal (could go either way). Always report thep-value so your readers can draw their own conclusions.

For example, suppose a pizza place claims their delivery times are 30 minutes or less on average but you think its more than that. You conduct a hypothesis test because you believe the null hypothesis, Ho, that the mean delivery time is 30 minutes max, is incorrect. Your alternative hypothesis (Ha) is that the mean time is greater than 30 minutes. You randomly sample some delivery times and run the data through the hypothesis test, and yourp-value turns out to be 0.001, which is much less than 0.05. In real terms, there is a probability of 0.001 that you will mistakenly reject the pizza places claim that their delivery time is less than or equal to 30 minutes. Since typically we are willing to reject the null hypothesis when this probability is less than 0.05, you conclude that the pizza place is wrong; their delivery times are in fact more than 30 minutes on average, and you want to know what theyre gonna do about it! (Of course, you could be wrong by having sampled an unusually high number of late pizza deliveries just by chance.)

In HYPERLINK "http://en.wikipedia.org/wiki/Statistical_inference" \o "Statistical inference" statistical inference of observed data of a HYPERLINK "http://en.wikipedia.org/wiki/Scientific_experiment" \o "Scientific experiment" scientific experiment, the null hypothesis refers to a general or default position: that there is no relationship between two measured phenomena, or that a potential medical treatment has no effect. Rejecting or disproving the null HYPERLINK "http://en.wikipedia.org/wiki/Hypothesis" \o "Hypothesis" hypothesis and thus concluding that there are grounds for believing that there is a relationship between two phenomena or that a potential treatment has a measurable effect is a central task in the modern practice of science, and gives a precise sense in which a claim is HYPERLINK "http://en.wikipedia.org/wiki/Falsifiability" \o "Falsifiability" capable of being proven false.