Question

for the following set of data, find the percentage of data within 1 population standard deviation of the mean, to the nearest 10th of a percent. 66, 63, 55, 64, 66, 75, 68, 64, 67

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{66 + 63+55+64+66+75+68+64+67}{9}=\frac{588}{9}\approx65.33$

Step2: Calculate the population standard - deviation

The formula for population standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$
$(66 - 65.33)^{2}=0.4489$, $(63 - 65.33)^{2}=5.4289$, $(55 - 65.33)^{2}=106.7089$, $(64 - 65.33)^{2}=1.7689$, $(66 - 65.33)^{2}=0.4489$, $(75 - 65.33)^{2}=93.5089$, $(68 - 65.33)^{2}=7.1289$, $(64 - 65.33)^{2}=1.7689$, $(67 - 65.33)^{2}=2.7889$
$\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=0.4489+5.4289+106.7089+1.7689+0.4489+93.5089+7.1289+1.7689+2.7889 = 219.9111$
$\sigma=\sqrt{\frac{219.9111}{9}}\approx\sqrt{24.4346}\approx4.94$

Step3: Determine the range

The range is $\bar{x}-\sigma$ to $\bar{x}+\sigma$, so $65.33 - 4.94=60.39$ and $65.33 + 4.94 = 70.27$

Step4: Count the number of data points in the range

The data points $63,64,66,64,66,68,67$ are in the range. There are 7 data - points.

Step5: Calculate the percentage

The percentage is $\frac{7}{9}\times100\%\approx77.8\%$

Answer:

$77.8\%$