I Really need HELP!!!!!

Question

Confirmed Measles Cases. The data show a sample of the number of confirmed measles cases over a recent 12-year period. Find the range, variance, and standard deviation for the data.

212 63 71 140 43 55 6 37 56 44 116 86

Answer 1

We're going to assume that this data is a population and not a sample. (the equations and methods change slightly if we are considering the 12 years as a sample of a larger population.)

Step 1) Sort the data, small to large. The range is the largest minus the smallest. Here is the data sorted:

6

37

43

44

55

56

63

71

86

116

140

212

So the range is 212 - 6 or 206

Step 2) To find the variance and standard deviation, it's easy if you have a calculator such as a TI-84 that lets you enter the data in a list then push a button to get the answer. I'm going to assume you do not have such a calculator and that you have to do it manually.

Build a table with the data values in column 1 (you already did this part but here it is again):

6

37

43

44

55

56

63

71

86

116

140

212

Calculate the mean (average) of the data by adding them together and dividing by the number of values (12). The sum of all the numbers is 929 and dividing by 12 gives 77.41667

Add a second column that subtracts the data value and the mean (average), so the first entry would be 6 - 77.41667 = -71.41667. The completed looks like this (note that I show rounded numbers for simplicity but I'm not actually rounding the numbers for the calculations)

Data (x) x - mean

6 -71.4

37 -40.4

43 -34.4

44 -33.4

55 -22.4

56 -21.4

63 -14.4

71 -6.4

86 8.6

116 38.6

140 62.6

212 134.6

Next, add a third column that squares the second column. So the first entry would be (-71.41667)² or 5100.34028. Here is the result (again with the numbers rounded for simplicity. Don't use the rounded numbers in the calculations because you'll introduce error and the answer will be off.

6 -71.4 5100.3

37 -40.4 1633.5

43 -34.4 1184.5

44 -33.4 1116.7

55 -22.4 502.5

56 -21.4 458.7

63 -14.4 207.8

71 -6.4 41.2

86 8.6 73.7

116 38.6 1488.7

140 62.6 3916.7

212 134.6 18112.7

Next add the 3rd column up and divide by the number of data values (12). The sum of the 3rd column is 33836.9167 and dividing by 12 gives 2819.74. That is the variance.

The standard deviation is the square root of the variance so √2819.74 so the standard deviation is 53.1