Question

use the data 6, 7, 8, 10, 11, 12. without actually computing the standard deviation, determine the value below that best approximates the standard deviation
choose the correct answer below.
a. 20
b. 6
c. 10
d. 2

Explanation:

Step1: Analyze data range

The data set is 6, 7, 8, 10, 11, 12. The minimum value is 6 and the maximum value is 12. The range is \( 12 - 6 = 6 \).

Step2: Relate range to standard deviation

For a small data set, the standard deviation is typically less than the range and often a reasonable fraction of it. The range here is 6, so the standard deviation should be less than 6 but not extremely small (like 2) or as large as 10 or 20. Among the options, 2 is too small, 10 and 20 are larger than the range, so the most reasonable approximation is around 2 - 3, but from the options, 2 is too low, wait, wait, maybe I miscalculated. Wait, the data points: 6,7,8,10,11,12. Let's check the spread. The mean is around (6+7+8+10+11+12)/6 = (54)/6 = 9. Then the deviations from the mean: -3, -2, -1, +1, +2, +3. The squared deviations: 9,4,1,1,4,9. The variance would be (9+4+1+1+4+9)/5 = 28/5 = 5.6, so standard deviation is \( \sqrt{5.6} \approx 2.37 \), which is close to 2? Wait no, wait the options: A.20, B.6, C.10, D.2. Wait, my calculation shows around 2.37, so the closest is D? Wait no, wait maybe I made a mistake. Wait the data is 6,7,8,10,11,12. The range is 6 (12-6). Standard deviation is a measure of spread, and for this data, the values are relatively close. Let's see the options: 20 is way too big, 10 too big, 6 is the range, but standard deviation is less than range. The actual standard deviation calculation: mean is 9. Deviations: -3, -2, -1, +1, +2, +3. Sum of squared deviations: 9+4+1+1+4+9=28. Variance: 28/5=5.6. Standard deviation: sqrt(5.6)≈2.37, so approximately 2. So the answer is D? Wait but earlier I thought maybe, but let's check again. So the correct approximation is around 2, so option D.

Answer:

D. 2