Q. 3.204 - Elementary Statistics

Question

We have provided simple data sets for you to practice the basics of finding

(a) Population mean

(b) Population standard deviation

Data is $1,9,8,4,3$

Step-by-Step Solution

Answer

(a) Population Mean$\left(\mu \right)$= $5$

(b) Population Standard deviation$\left(\sigma \right)$= $3.033$

Part(a) Step 1: Given information

We are given simple data set of $1,9,8,4,3$

We need to find population mean of given data 

Part(a) Step 2: Simplify

A population in statistics is a group of similar items or events that relate to a query or experiment.

Population Mean$\left(\mu \right)$=$\frac{\sum _{} x_i}{N}$

$x_i$ denotes individual value

$N$ denotes total population

$$\begin{gathered} \mu =\frac{1+9+8+4+3}{5} \\ \mu =\frac{25}{5}=5 \end{gathered}$$

Hence, mean of given data is $5$ 

Part (b) Step 1:Given Information

We are provided simple data having $1,9,8,4,3$

We need to find standard deviation of given population.

Part(b) Step 2: Simplify

We know standard deviation is square root of variance.

Formula of variance is 

${\sigma }^2=\frac{{(x_1-\mu )}^2+...........+{(x_N-\mu )}^2}{N}$

Where $x_i$ denotes $i_{th}$ value

$\mu$ denotes population mean

$N$ denotes total population

Using above formula, Standard deviation is 

$$\begin{gathered} \sigma =\sqrt{\frac{{(1-5)}^2+{(9-5)}^2+{(8-5)}^2+{(4-5)}^2+{(3-5)}^2}{5}} \\ =\sqrt{\frac{4^2+4^2+3^2+1^2+2^2}{5}} \\ =3.033 \end{gathered}$$

Hence, population standard deviation of given data is $3.033$