Question

for the following set of data, find the sample standard deviation, to the nearest thousandth.

data	frequency
10	1
11	3
13	3
15	8
16	5
17	1
19	4
20	9

copy values for calculator
open statistics calculator

Explanation:

Step1: Calculate the sample mean $\bar{x}$

First, find the product of each data - value and its frequency:
For $x = 5$ and $f = 6$, the product is $5\times6=30$.
For $x = 10$ and $f = 1$, the product is $10\times1 = 10$.
For $x = 11$ and $f = 3$, the product is $11\times3=33$.
For $x = 13$ and $f = 3$, the product is $13\times3 = 39$.
For $x = 15$ and $f = 8$, the product is $15\times8=120$.
For $x = 16$ and $f = 5$, the product is $16\times5 = 80$.
For $x = 17$ and $f = 1$, the product is $17\times1=17$.
For $x = 19$ and $f = 4$, the product is $19\times4 = 76$.
For $x = 20$ and $f = 9$, the product is $20\times9=180$.

The sum of the products $\sum_{i = 1}^{n}x_if_i=30 + 10+33+39+120+80+17+76+180=585$.

The total frequency $n=\sum_{i = 1}^{n}f_i=6 + 1+3+3+8+5+1+4+9=40$.

The sample mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_if_i}{n}=\frac{585}{40}=14.625$.

Step2: Calculate the squared - differences and their weighted sum

For $x = 5$: $(5 - 14.625)^2\times6=( - 9.625)^2\times6=92.640625\times6 = 555.84375$.
For $x = 10$: $(10 - 14.625)^2\times1=( - 4.625)^2\times1 = 21.390625$.
For $x = 11$: $(11 - 14.625)^2\times3=( - 3.625)^2\times3=13.140625\times3 = 39.421875$.
For $x = 13$: $(13 - 14.625)^2\times3=( - 1.625)^2\times3=2.640625\times3 = 7.921875$.
For $x = 15$: $(15 - 14.625)^2\times8=(0.375)^2\times8 = 0.140625\times8=1.125$.
For $x = 16$: $(16 - 14.625)^2\times5=(1.375)^2\times5 = 1.890625\times5=9.453125$.
For $x = 17$: $(17 - 14.625)^2\times1=(2.375)^2\times1 = 5.640625$.
For $x = 19$: $(19 - 14.625)^2\times4=(4.375)^2\times4 = 19.140625\times4=76.5625$.
For $x = 20$: $(20 - 14.625)^2\times9=(5.375)^2\times9 = 28.890625\times9=260.015625$.

The sum of the weighted squared - differences $\sum_{i = 1}^{n}f_i(x_i-\bar{x})^2=555.84375+21.390625+39.421875+7.921875+1.125+9.453125+5.640625+76.5625+260.015625 = 977.375$.

Step3: Calculate the sample standard deviation $s$

The formula for the sample standard deviation is $s=\sqrt{\frac{\sum_{i = 1}^{n}f_i(x_i - \bar{x})^2}{n - 1}}$.
Since $n = 40$, then $s=\sqrt{\frac{977.375}{39}}\approx\sqrt{25.060897}\approx5.006$.

Answer:

$5.006$