236,031 views
7 votes
7 votes
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

For the following set of data, find the number of data within 2 population standarddeviations-example-1
User Munesh
by
2.4k points

1 Answer

23 votes
23 votes

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.

User Jonathan Musso
by
3.0k points