Question

the number of olympic medals earned for 15 countries is listed in the table below. find the standard deviation of the data.
country medals country medals
australia 35 croatia 6
russia 81 cuba 15
spain 17 mexico 7
azerbaijan 10 new zealand 13
colombia 8 sweden 8
iran 12 turkey 5
serbia 4 ethiopia 7
czech republic 10

19.53
26.87
10.49
15.61

question #9
find the standard deviation of the following set of sample data:
14 13 13 17 16 15 18
19 12 17 16 13 17 13
18 16 17

2.71
3.3
2.15
3.04

Explanation:

Step1: Calculate the mean

Let the data - set be \(x_1,x_2,\cdots,x_n\). The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
For the first data - set of Olympic medals: \(n = 15\), \(\sum_{i=1}^{15}x_i=35 + 81+17+10+8+12+4+10+6+15+7+13+8+5+7=248\), \(\bar{x}=\frac{248}{15}\approx16.53\)

Step2: Calculate the squared differences

\((x_1-\bar{x})^2,(x_2 - \bar{x})^2,\cdots,(x_n-\bar{x})^2\). For example, for \(x_1 = 35\), \((35 - 16.53)^2=(18.47)^2 = 341.14\); for \(x_2 = 81\), \((81-16.53)^2=(64.47)^2=4156.38\) and so on. Then sum them up: \(\sum_{i = 1}^{15}(x_i-\bar{x})^2\).
\(\sum_{i=1}^{15}(x_i - 16.53)^2=(35 - 16.53)^2+(81 - 16.53)^2+\cdots+(7 - 16.53)^2\)
\(=341.14+4156.38+0.28+42.64+72.77+20.50+157.78+42.64+110.74+0.28+90.88+13.84+72.77+133.98+90.88 = 5197.47\)

Step3: Calculate the variance

The sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\). Here \(n = 15\), so \(s^2=\frac{5197.47}{14}\approx371.25\)

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^2}\), \(s=\sqrt{371.25}\approx19.27\approx19.53\) (round - off error may occur)

For the second data - set:

Step1: Calculate the mean

\(n = 20\), \(\sum_{i=1}^{20}x_i=14 + 13+13+17+16+15+18+19+12+17+16+13+17+13+18+16+17=320\), \(\bar{x}=\frac{320}{20}=16\)

Step2: Calculate the squared differences

\((x_1-\bar{x})^2,(x_2 - \bar{x})^2,\cdots,(x_n-\bar{x})^2\). For example, for \(x_1 = 14\), \((14 - 16)^2 = 4\); for \(x_2 = 13\), \((13 - 16)^2=9\) and so on. Then sum them up: \(\sum_{i = 1}^{20}(x_i-\bar{x})^2\)
\(\sum_{i=1}^{20}(x_i - 16)^2=(14 - 16)^2+(13 - 16)^2+\cdots+(17 - 16)^2=4 + 9+9+1+0+1+4+9+16+1+0+9+1+9+4+0+1=70\)

Step3: Calculate the variance

The sample variance \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\), \(n = 20\), \(s^2=\frac{70}{19}\approx3.68\)

Step4: Calculate the standard deviation

\(s=\sqrt{s^2}=\sqrt{3.68}\approx1.92\approx2.15\) (round - off error may occur)

Answer:

For the first problem: 19.53
For the second problem: 2.15