PART VIII. TESTS OF SIGNIFICANCE
5. THE STANDARD DEVIATION
As the quote at the beginning of the chapter suggests, it is often helpful to think of the way a list of numbers spreads out around the average. This spread is usually measured by a quantity called thestandard deviation, or SD. The SD measures the size of deviations from the average: it is a sort of average deviation.
The program is to interpret the SD in the context of real data, and then see how to calculate it.
There were 2,696 women age 18 and over in the HANES5 sample. The av- erage height of these women was about 63.5 inches, and the SD was close to 3 inches. The average tells us that most of the women were somewhere around 63.5 inches tall. But there were deviations from the average. Some of the women were taller than average, some shorter. How big were these deviations? That is where the SD comes in.
The SD says how far away numbers on a list are from their av- erage. Most entries on the list will be somewhere around one SD away from the average. Very few will be more than two or three SDs away.
The SD of 3 inches says that many of the women differed from the average height by 1 or 2 or 3 inches: 1 inch is a third of an SD, and 3 inches is an SD. Few women differed from the average height by more than 6 inches (two SDs).
There is a rule of thumb which makes this idea more quantitative, and which applies to many data sets.
Roughly 68% of the entries on a list (two in three) are within one SD of the average, the other 32% are further away. Roughly 95%
(19 in 20) are within two SDs of the average, the other 5% are further away. This is so for many lists, but not all.
Figure 8 shows the histogram for the heights of women age 18 and over in HANES5. The average is marked by a vertical line, and the region within one SD of the average is shaded. This shaded area represents the women who differed from average height by one SD or less. The area is about 72%. About 72% of the women differed from the average height by one SD or less.
Figure 8. The SD and the histogram. Heights of 2,696 women age 18 and over in HANES5. The average of 63.5 inches is marked by a vertical line. The region within one SD of the average is shaded: 72% of the women differed from average by one SD (3 inches) or less.
0 10 20
54 56 58 60 62 64 66 68 70 72 74
HEIGHT (INCHES)
PERCENT PER INCH
THE STANDARD DEVIATION 69
Figure 9 shows the same histogram. Now the area within two SDs of average is shaded. This shaded area represents the women who differed from average height by two SDs or less. The area is about 97%. About 97% of the women differed from the average height by two SDs or less.
Figure 9. The SD and the histogram. Heights of 2,696 women age 18 and over in HANES5. The average of 63.5 inches is marked by a vertical line. The region within two SDs of the average is shaded: 97% of the women differed from average by two SDs (6 inches) or less.
0 10 20
54 56 58 60 62 64 66 68 70 72 74
HEIGHT (INCHES)
PERCENT PER INCH
To sum up, about 72% of the women differed from average by one SD or less, and 97% differed from average by two SDs or less. There was only one woman in the sample who was more than three SDs away from the average, and none were more than four SDs away. For this data set, the 68%–95% rule works quite well.
Where do the 68% and 95% come from? See chapter 5.10
About two-thirds of the HANES women differed from the average by less than one SD.
Exercise Set D
1. The Public Health Service found that for boys age 11 in HANES2, the average height was 146 cm and the SD was 8 cm. Fill in the blanks.
(a) One boy was 170 cm tall. He was above average, by SDs.
(b) Another boy was 148 cm tall. He was above average, by SDs.
(c) A third boy was 1.5 SDs below average height. He was cm tall.
(d) If a boy was within 2.25 SDs of average height, the shortest he could have been is cm and the tallest is cm.
2. This continues exercise 1.
(a) Here are the heights of four boys: 150 cm, 130 cm, 165 cm, 140 cm. Match the heights with the descriptions. A description may be used twice.
unusually short about average unusually tall
(b) About what percentage of boys age 11 in the study had heights between 138 cm and 154 cm? Between 130 and 162 cm?
3. Each of the following lists has an average of 50. For which one is the spread of the numbers around the average biggest? smallest?
(i) 0, 20, 40, 50, 60, 80, 100 (ii) 0, 48, 49, 50, 51, 52, 100 (iii) 0, 1, 2, 50, 98, 99, 100
4. Each of the following lists has an average of 50. For each one, guess whether the SD is around 1, 2, or 10. (This does not require any arithmetic.)
(a) 49, 51, 49, 51, 49, 51, 49, 51, 49, 51 (b) 48, 52, 48, 52, 48, 52, 48, 52, 48, 52 (c) 48, 51, 49, 52, 47, 52, 46, 51, 53, 51 (d) 54, 49, 46, 49, 51, 53, 50, 50, 49, 49 (e) 60, 36, 31, 50, 48, 50, 54, 56, 62, 53
5. The SD for the ages of the people in the HANES5 sample is around . Fill in the blank, using one of the options below. Explain briefly. (This survey was discussed in section 2; the age range was 0–85 years.)
5 years 25 years 50 years
6. Below are sketches of histograms for three lists. Match the sketch with the de- scription. Some descriptions will be left over. Give your reasoning in each case.
(i) ave≈3.5, SD≈1 (iv) ave≈2.5, SD≈1 (ii) ave≈3.5, SD≈0.5 (v) ave≈2.5, SD≈0.5 (iii) ave≈3.5, SD≈2 (vi) ave≈4.5, SD≈0.5
COMPUTING THE STANDARD DEVIATION 71
7. (Hypothetical). In a clinical trial, data collection usually starts at “baseline,” when the subjects are recruited into the trial but before they are randomized to treatment or control. Data collection continues until the end of followup. Two clinical trials on prevention of heart attacks report baseline data on weight, shown below. In one of these trials, the randomization did not work. Which one, and why?
Number of Average
persons weight SD
Treatment 1,012 185 lb 25 lb
(i)
!
Control 997 143 lb 26 lb
Treatment 995 166 lb 27 lb
(ii)
!
Control 1,017 163 lb 25 lb
8. One investigator takes a sample of 100 men age 18–24 in a certain town. Another takes a sample of 1,000 such men.
(a) Which investigator will get a bigger average for the heights of the men in his sample? or should the averages be about the same?
(b) Which investigator will get a bigger SD for the heights of the men in his sample? or should the SDs be about the same?
(c) Which investigator is likely to get the tallest of the sample men? or are the chances about the same for both investigators?
(d) Which investigator is likely to get the shortest of the sample men? or are the chances about the same for both investigators?
9. The men in the HANES5 sample had an average height of 69 inches, and the SD was 3 inches. Tomorrow, one of these men will be chosen at random. You have to guess his height. What should you guess? You have about 1 chance in 3 to be off by more than . Fill in the blank. Options: 1/2 inch, 3 inches, 5 inches.
10. As in exercise 9, but tomorrow a whole series of men will be chosen at random.
After each man appears, his actual height will be compared with your guess to see how far off you were. The r.m.s. size of the amounts off should be . Fill in the blank. (Hint: Look at the bottom of this page.)
The answers to these exercises are on pp. A49–50.