Question

question
for the following set of data, find the number of data within 2 population standard deviations of the mean.
105, 108, 65, 57, 64, 62, 64, 66
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Explanation:

Step1: Calculate the population mean ($\mu$)

First, we find the sum of the data set: \(105 + 108 + 65 + 57 + 64 + 62 + 64 + 66\)

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The number of data points (\(N\)) is 8. So the mean \(\mu=\frac{591}{8}=73.875\)

Step2: Calculate the population standard deviation ($\sigma$)

The formula for population standard deviation is \(\sigma=\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}\)
First, we calculate \((x_{i}-\mu)^{2}\) for each data point:

• For \(x = 105\): \((105 - 73.875)^{2}=(31.125)^{2}=968.765625\)
• For \(x = 108\): \((108 - 73.875)^{2}=(34.125)^{2}=1164.515625\)
• For \(x = 65\): \((65 - 73.875)^{2}=(-8.875)^{2}=78.765625\)
• For \(x = 57\): \((57 - 73.875)^{2}=(-16.875)^{2}=284.765625\)
• For \(x = 64\): \((64 - 73.875)^{2}=(-9.875)^{2}=97.515625\) (we have two data points with \(x = 64\), so we calculate this twice)
• For \(x = 62\): \((62 - 73.875)^{2}=(-11.875)^{2}=141.015625\)
• For \(x = 66\): \((66 - 73.875)^{2}=(-7.875)^{2}=62.015625\)

Now, sum up these squared differences:

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Now, divide by \(N = 8\): \(\frac{2894.875}{8}=361.859375\)
Then take the square root: \(\sigma=\sqrt{361.859375}\approx19.0226\)

Step3: Find the range within \(2\sigma\) of the mean

The lower bound is \(\mu - 2\sigma=73.875-2\times19.0226=73.875 - 38.0452 = 35.8298\)
The upper bound is \(\mu + 2\sigma=73.875+2\times19.0226=73.875 + 38.0452 = 111.9202\)

Step4: Count the data points within the range

Now we check each data point:

• \(105\): \(35.8298<105<111.9202\) (yes)
• \(108\): \(35.8298<108<111.9202\) (yes)
• \(65\): \(35.8298<65<111.9202\) (yes)
• \(57\): \(35.8298<57<111.9202\) (yes)
• \(64\): \(35.8298<64<111.9202\) (yes)
• \(62\): \(35.8298<62<111.9202\) (yes)
• \(64\): \(35.8298<64<111.9202\) (yes)
• \(66\): \(35.8298<66<111.9202\) (yes)

All 8 data points are within \(2\) population standard deviations of the mean.

Answer:

8