Question

For the following set of data, find the number of data within 2 population standarddeviations of the mean.93, 65, 67, 67, 67, 65, 66, 65

Answer 1

First we need to find the mean and the standadr deviation. For the mean, we sum the values up and divide by the number of data given. We have 8 data, so:

`\mu=(93+65+67+67+67+65+66+65)/(8)=(555)/(8)=69.375`

Now, we need the standard deviation. To find it, we use the formula:

`\sigma=\sqrt[]{\frac{\sum ^{}_{}(x_i-\mu)^2}{N-1}}`

The sum symbol in there mean we need to get each data given, substract the mean we calculated, square them and then sum all. N is the number of data, which is 8. Let look one example:

For data 93, we do:

`(x_i-\mu)^2=(93-69.375)^2=23.625^2=558.141`

Now, we do the same for the others:

`\begin{gathered} (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(67-69.375)^2=(-2.375)^2=5.641 \\ (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \\ (x_i-\mu)^2=(66-69.375)^2=(-3.375)^2=11.391 \\ (x_i-\mu)^2=(65-69.375)^2=(-4.375)^2=19.141 \end{gathered}`

Now we sum all of them:

`558.141+19.141+5.641+5.641+5.641+19.141+11.391+19.141=643.878`

This goes into the formula.

`\sigma=\sqrt[]{(643.878)/(8-1)}=\sqrt[]{(643.878)/(7)}=\sqrt[]{91.982}=9.591`

Now, we want the number of data that is within 2 standard deviation from the mean. Thus, we want the data that are between:

`\begin{gathered} \mu-2\sigma=69.375-2\cdot9.591=50.193 \\ \mu+2\sigma=69.375+2\cdot9.591=88.557 \end{gathered}`

From the given Data, 93 is out of this range, but all the others (65, 67, 67, 67, 65, 66, 65) are in it. So, the number of data within 2 standard deviations from the mean is 7.