Question

a. Calculate the sample average using ungrouped and grouped methods.

b. Calculate the sample standard deviation using ungrouped and grouped method

Answer 1

(a-b)

For un-grouped data:

Following table shows the calculations:

	X	(x-mean)^2
	10	17.0569
	7	50.8369
	12	4.5369
	11	9.7969
	9	26.3169
	13	1.2769
	14	0.0169
	10	17.0569
	6	66.0969
	12	4.5369
	17	8.2369
	17	8.2369
	13	1.2769
	15	0.7569
	18	14.9769
	14	0.0169
	15	0.7569
	16	3.4969
	15	0.7569
	14	0.0169
	9	26.3169
	19	23.7169
	15	0.7569
	14	0.0169
	15	0.7569
	16	3.4969
	15	0.7569
	14	0.0169
	13	1.2769
	16	3.4969
	17	8.2369
	13	1.2769
	14	0.0169
	11	9.7969
	12	4.5369
	15	0.7569
	16	3.4969
	13	1.2769
	16	3.4969
	15	0.7569
	18	14.9769
	15	0.7569
	13	1.2769
	15	0.7569
	14	0.0169
	16	3.4969
	13	1.2769
	16	3.4969
	15	0.7569
	16	3.4969
	20	34.4569
	14	0.0169
	10	17.0569
	15	0.7569
	13	1.2769
	15	0.7569
	12	4.5369
	14	0.0169
	16	3.4969
	13	1.2769
	16	3.4969
	13	1.2769
	14	0.0169
	16	3.4969
	14	0.0169
	15	0.7569
	16	3.4969
	15	0.7569
	16	3.4969
	15	0.7569
Total	989	441.843
	14.12857	2.530518077

Sample size: n=70

The sample mean is:

$\bar{x}=\frac{\sum x}{n}=\frac{989}{70}=14.13$

The sample standard deviation is

$s=\sqrt{\frac{\sum \left ( x-\bar{x} \right )^{2}}{n-1}}=\sqrt{\frac{441.843}{70-1}}\approx 2.53$

For grouped data:

Following table shows the calculations:

Classes
Lower limit	Upper limit	Mid Point, X	Frequency, f	fx	f(x-mean)^2
6	8.1	7.05	2	14.1	97.7202
8.1	10.2	9.15	5	45.75	119.5605
10.2	12.3	11.25	6	67.5	46.7046
12.3	14.4	13.35	21	280.35	9.9981
14.4	16.5	15.45	29	448.05	57.6549
16.5	18.6	17.55	5	87.75	61.6005
18.6	20.7	19.65	2	39.3	62.9442
Total			70	982.8	456.183

The sample mean is

$\bar{x}=\frac{\sum fx}{\sum f}=14.04$

The sample standard deviation is

$s=\sqrt{\frac{\sum f(x-\bar{x})^{2}}{\sum f -1}}=\sqrt{\frac{456.183}{69}}=2.57$