Question

question for the following set of data, find the population standard deviation, to the nearest thousandth. 71, 27, 57, 15, 92, 66, 85 answer attempt 1 out of 2 open statistics calculator copy values for calculator watch video show examples

Explanation:

Step1: Calculate the mean

Let the data - set be \(x_1 = 71,x_2 = 27,x_3 = 57,x_4 = 15,x_5 = 92,x_6 = 66,x_7 = 85\). The mean \(\mu=\frac{\sum_{i = 1}^{n}x_i}{n}\), where \(n = 7\).
\(\sum_{i=1}^{7}x_i=71 + 27+57 + 15+92+66+85=413\), so \(\mu=\frac{413}{7}=59\).

Step2: Calculate the squared - differences

\((x_1-\mu)^2=(71 - 59)^2=12^2 = 144\), \((x_2-\mu)^2=(27 - 59)^2=(-32)^2 = 1024\), \((x_3-\mu)^2=(57 - 59)^2=(-2)^2 = 4\), \((x_4-\mu)^2=(15 - 59)^2=(-44)^2 = 1936\), \((x_5-\mu)^2=(92 - 59)^2=33^2 = 1089\), \((x_6-\mu)^2=(66 - 59)^2=7^2 = 49\), \((x_7-\mu)^2=(85 - 59)^2=26^2 = 676\).

Step3: Calculate the variance

The population variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\mu)^2}{n}\).
\(\sum_{i = 1}^{7}(x_i-\mu)^2=144 + 1024+4+1936+1089+49+676=4922\), so \(\sigma^{2}=\frac{4922}{7}=703.1429\).

Step4: Calculate the standard deviation

The population standard deviation \(\sigma=\sqrt{\sigma^{2}}\).
\(\sigma=\sqrt{703.1429}\approx26.517\)

Answer:

\(26.517\)