Question

course: probability and statistics
consider the data below, calculate the mean and standard for grouped data.
1 3 1 1 1
1 2 2 1 0
1 1 0 0 0
1 5 0 3 6
3 0 3 1 1
3 6 6 1 1
the table is from the data

x	f
0	5
1	13
2	2
3	5
5	1
6	3

Explanation:

Step1: Calculate the mid - points

Let the data values be $x$ and frequencies be $f$. We assume the data is already grouped in a simple way where $x$ represents the mid - points of the groups.

Step2: Calculate the product $xf$

$x$	$f$	$xf$
1	13	$1\times13=13$
2	2	$2\times2 = 4$
3	5	$3\times5=15$
5	1	$5\times1 = 5$
6	3	$6\times3=18$

The sum of frequencies $\sum f=5 + 13+2+5+1+3=29$.
The sum of $xf$ values $\sum xf=0 + 13+4+15+5+18=55$.

Step3: Calculate the mean $\bar{x}$

The mean formula for grouped data is $\bar{x}=\frac{\sum xf}{\sum f}$.
$\bar{x}=\frac{55}{29}\approx1.9$.

Step4: Calculate the product $(x - \bar{x})^2f$

$x$	$f$	$(x - \bar{x})^2$	$(x - \bar{x})^2f$
1	13	$(1 - 1.9)^2=0.81$	$0.81\times13 = 10.53$
2	2	$(2 - 1.9)^2=0.01$	$0.01\times2=0.02$
3	5	$(3 - 1.9)^2 = 1.21$	$1.21\times5=6.05$
5	1	$(5 - 1.9)^2=9.61$	$9.61\times1 = 9.61$
6	3	$(6 - 1.9)^2=16.81$	$16.81\times3=50.43$

The sum of $(x - \bar{x})^2f$ is $18.05+10.53+0.02+6.05+9.61+50.43=94.69$.

Step5: Calculate the standard deviation $s$

The formula for the standard deviation of grouped data is $s=\sqrt{\frac{\sum(x - \bar{x})^2f}{\sum f- 1}}$.
$s=\sqrt{\frac{94.69}{29 - 1}}=\sqrt{\frac{94.69}{28}}\approx\sqrt{3.382}\approx1.84$.

Answer:

Mean: $\frac{55}{29}\approx1.9$, Standard deviation: $\sqrt{\frac{94.69}{28}}\approx1.84$