Question

1. Find the mean, mod, median, and standard deviation of the following data: 5, 6, 7, 8, 9,8,7,8 Based on these results, check whether the value of 10 is usual?

Answer 1

Calculation for Mean, Median , Mode and Standard deviation for the given data:

Let the data set be

x = {5, 6, 7, 8, 9, 8, 7,8

Mean: The mean is calculated for the given data set as

Σε γ = η.

where, 'n' is the number of data in the given data set, in this case n=8

Σx 5 +6 +7+8 +9 +8+7+8 58 Ξs = 7. 25

So, the mean is calculated as

X= 7.25

Median: The median is the middle value, and it is a measure of the central tendency. As it represents the value for below which 50% data present and above which 50% of the data present.

To find median we first need to arrange the data in ordered way, i.e.,

x = {5,6,7,7,8,8,8,9

When the number of data is EVEN the median is calculated as the average of

th 01) and (65 +1term

Since, $\mathrm{n=8}$ which is even.

Then,   

= 4th term NO la and the 4th term = 7

th +1) = 5th term and the 5th term = 8

So, the median is the average of term.

14th + 5th |

$\mathrm{Median=\left ( \frac{4^{th}+5^{th}}{2} \right )=\left ( \frac{7+8}{2} \right )=\frac{15}{2}=\mathbf{7.5}}$

Hence, the Median is calculated as 7.5

Mode: It is the value that occurs the most in the data set.

For the given data set, i.e., the value 8 occurs three times in the data set, so it'll be the mode.

x = {5, 6, 7, 8, 9, 8, 7,8

So, the Mode=8

Standard deviation $\mathrm{(\sigma):}$

Standard deviation is calculated as- $\mathrm{\sigma=\sqrt{\frac{\sum{(x_{i}-\bar{x})^{2}}}{n}}}$

We have already calculated the mean, i.e., $\mathrm{\bar{x}=7.25}$

$\mathrm{x}$	$\mathrm{(x-\bar{x})}$	$\mathrm{(x-\bar{x})^{2}}$
5	5-7.25= -2.25	5.0625
6	6-7.25= -1.25	1.5625
7	7-7.25= -0.25	0.0625
8	8-7.25= 0.75	0.5625
9	9-7.25= 1.75	3.0625
8	8-7.25= 0.75	0.5625
7	7-7.25= -0.25	0.0625
8	8-7.25= 0.75	0.5625
		$\mathrm{(\mathbf{x-\bar{x})^{2}=11.5}}$

71 2860861'1 – cü 18193--

So, the standard deviation calculated as $\mathrm{\sigma= \mathbf{1.2}}$ (round to one decimal place)

Based on these results the value 10 is within the s standard deviation of mean, i.e., $\bar{x}+3\sigma=7.25+3*1.2=10.85$ , so the value 10 is within the three standard deviation, so, we can say that the value 10 is usual.