Question

for the following set of data, find the number of data within 2 population standard deviations of the mean.

data	frequency
14	7
17	11
22	12
24	12
25	8
28	2

Explanation:

Step1: Calculate the mean $\mu$

First, find the sum of the products of data - value and frequency $\sum_{i = 1}^{n}x_if_i$ and the total frequency $\sum_{i = 1}^{n}f_i$.
Let $x_i$ be the data - value and $f_i$ be the frequency.
$\sum_{i = 1}^{n}x_if_i=13\times2 + 14\times7+17\times11 + 22\times12+24\times12+25\times8+28\times2$
$=26+98 + 187+264+288+200+56$
$=1119$.
$\sum_{i = 1}^{n}f_i=2 + 7+11+12+12+8+2=54$.
The mean $\mu=\frac{\sum_{i = 1}^{n}x_if_i}{\sum_{i = 1}^{n}f_i}=\frac{1119}{54}\approx20.72$.

Step2: Calculate the population variance $\sigma^{2}$

The formula for the population variance is $\sigma^{2}=\frac{\sum_{i = 1}^{n}f_i(x_i-\mu)^{2}}{\sum_{i = 1}^{n}f_i}$.
$(13 - 20.72)^2\times2=( - 7.72)^2\times2 = 59.5984\times2 = 119.1968$.
$(14 - 20.72)^2\times7=( - 6.72)^2\times7 = 45.1584\times7 = 316.1088$.
$(17 - 20.72)^2\times11=( - 3.72)^2\times11 = 13.8384\times11 = 152.2224$.
$(22 - 20.72)^2\times12=(1.28)^2\times12 = 1.6384\times12 = 19.6608$.
$(24 - 20.72)^2\times12=(3.28)^2\times12 = 10.7584\times12 = 129.1008$.
$(25 - 20.72)^2\times8=(4.28)^2\times8 = 18.3184\times8 = 146.5472$.
$(28 - 20.72)^2\times2=(7.28)^2\times2 = 52.9984\times2 = 105.9968$.
$\sum_{i = 1}^{n}f_i(x_i - \mu)^2=119.1968+316.1088+152.2224+19.6608+129.1008+146.5472+105.9968 = 988.8336$.
$\sigma^{2}=\frac{988.8336}{54}\approx18.31$.
The population standard - deviation $\sigma=\sqrt{18.31}\approx4.28$.

Step3: Find the range within 2 population standard - deviations of the mean

The lower limit is $\mu - 2\sigma=20.72-2\times4.28=20.72 - 8.56 = 12.16$.
The upper limit is $\mu + 2\sigma=20.72+2\times4.28=20.72 + 8.56 = 29.28$.

Step4: Count the number of data within the range

For $x = 13$, frequency $f = 2$.
For $x = 14$, frequency $f = 7$.
For $x = 17$, frequency $f = 11$.
For $x = 22$, frequency $f = 12$.
For $x = 24$, frequency $f = 12$.
For $x = 25$, frequency $f = 8$.
For $x = 28$, frequency $f = 2$.
The total number of data within 2 population standard - deviations of the mean is $2 + 7+11+12+12+8+2=54$.

Answer:

54