Question

question for the following set of data, find the percentage of data within 2 population standard deviations of the mean, to the nearest percent. 58, 5, 109, 62, 65, 58, 62, 53, 65

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{58 + 5+109+62+65+58+62+53+65}{9}=\frac{537}{9}\approx59.67$.

Step2: Calculate the population - standard deviation

First, find the squared - differences from the mean for each data point.
$(58 - 59.67)^2\approx2.79$, $(5 - 59.67)^2\approx2984.49$, $(109 - 59.67)^2\approx2433.45$, $(62 - 59.67)^2\approx5.43$, $(65 - 59.67)^2\approx28.41$, $(58 - 59.67)^2\approx2.79$, $(62 - 59.67)^2\approx5.43$, $(53 - 59.67)^2\approx44.49$, $(65 - 59.67)^2\approx28.41$.
The sum of squared - differences $S=\sum_{i = 1}^{9}(x_i-\bar{x})^2=2.79+2984.49+2433.45+5.43+28.41+2.79+5.43+44.49+28.41 = 5535.6$.
The population standard deviation $\sigma=\sqrt{\frac{S}{n}}=\sqrt{\frac{5535.6}{9}}\approx\sqrt{615.07}\approx24.8$.

Step3: Find the range within 2 standard deviations of the mean

The lower bound $L=\bar{x}-2\sigma\approx59.67-2\times24.8 = 59.67 - 49.6=10.07$.
The upper bound $U=\bar{x}+2\sigma\approx59.67+2\times24.8 = 59.67 + 49.6 = 109.27$.

Step4: Count the number of data points within the range

The data points 58, 62, 65, 58, 62, 53, 65 are within the range. There are 7 data points out of 9.

Step5: Calculate the percentage

The percentage $P=\frac{7}{9}\times100\%\approx78\%$.

Answer:

78%