Question

: For the given data

38, 11, 40, 0, 26, 15, 5, 45, 7, 32, 2, 18, 42, 8, 31, 27, 4, 12, 35, 15, 0, 7, 28, 46, 9, 16, 29, 34, 10, 7, 5, 1, 17, 22, 35, 8, 36, 47, 11, 30, 19, 0, 16, 14, 16, 18, 41, 38, 2, 17, 42, 45, 48, 28, 7, 21, 8, 28, 5, 20.

Find the following (without using SPSS- Show all the steps -if necessary use spss to verify your answer) :

Mean, median, range, Q1, Q2, outliers if any, dot plot, 10% trimmed mean, , variance and S.D and Also construct a stem-and-leaf plot.

Answer 1

Observation table -

Sr. No.	x	(x-x_bar)^2
1	0	429.871
2	0	429.871
3	0	429.871
4	1	389.4043
5	2	350.9377
6	2	350.9377
7	4	280.0043
8	5	247.5377
9	5	247.5377
10	5	247.5377
11	7	188.6044
12	7	188.6044
13	7	188.6044
14	7	188.6044
15	8	162.1377
16	8	162.1377
17	8	162.1377
18	9	137.671
19	10	115.2044
20	11	94.73771
21	11	94.73771
22	12	76.27105
23	14	45.33773
24	15	32.87107
25	15	32.87107
26	16	22.40441
27	16	22.40441
28	16	22.40441
29	17	13.93775
30	17	13.93775
31	18	7.471093
32	18	7.471093
33	19	3.004433
34	20	0.537773
35	21	0.071113
36	22	1.604453
37	26	27.73781
38	27	39.27115
39	28	52.80449
40	28	52.80449
41	28	52.80449
42	29	68.33783
43	30	85.87117
44	31	105.4045
45	32	126.9379
46	34	176.0045
47	35	203.5379
48	35	203.5379
49	36	233.0712
50	38	298.1379
51	38	298.1379
52	40	371.2046
53	41	410.7379
54	42	452.2713
55	42	452.2713
56	45	588.8713
57	45	588.8713
58	46	638.4046
59	47	689.938
60	48	743.4713
Total	1244	12349.73

1) -

Mean for the given data is 20.7333.

2) -

Median = Average of (n/2)th & (n+2)/2th items ....... (since, n is even)

= Average of (60/2)th & (60+2)/2th items

= Average of 30th & 31th items

= (17+18)/2

Median = 17.5

Median for the given data is 17.5.

3) -

Range = Maximum value - Minimum value

Here, maximum value = 48 & minimum value = 0

So,  Range = Maximum value - Minimum value = 48 - 0 = 48

Range for the given data is 48.

4) -

Q1 = size of (n+1)/4th item

= size of (60+1)/4th item

= size of 15.25th item

= size of 15th item + 0.25(size of 16th item - size of 15th item)

= 8 + 0.25(8-8)

Q1 = 8

First quartile for the given data is 8.

5) -

Q3 = size of 3(n+1)/4th item

= size of 3(60+1)/4th item

= size of 45.75th item

= size of 45th item + 0.75(size of 46th item - size of 45th item)

= 32 + 0.75(34-32)

= 32 + 1.5

Q3 = 33.5

Third quartile for the given data is 33.5.

6) -

Here, IQR = Q3 - Q1 = 33.5 - 8 = 25.5

Q1 - 1.5(IQR) = 8 - 1.5(25.5) = 8 - 38.25 = -30.25

Q3 + 1.5(IQR) = 33.5 + 1.5(25.5) = 33.5 + 38.25 = 71.75

By IQR rule, if any number less than Q1 - 1.5(IQR) or greatre than Q3 + 1.5(IQR), then that number is an outlier.

But, in given data, no value is less than Q1 - 1.5(IQR) or greatre than Q3 + 1.5(IQR), so there is no any outlier in the data.

7) -

Dot plot -

For plotting dotplot, take values in the data to the axis then count each value in the data repeated for number of times. Plot a dot for each count of value in the data on y-axis corresponding to that number.

8) -

10% trimmed mean -

We want to calculate 10% trimmed mean, then the mean of the observations by excluding 10% smallest & 10% largest observations is known as 10% trimmed mean.

So, first we have to calculate P₁₀ & P₉₀ -

P₁₀ = Size of 10(n+1)/100th item

= Size of 10(60+1)/100th item

= Size of 6.1th item

= Size of 6th item + 0.1(size of 7th item - size of 6th item)

= 2 + 0.1(4-2)

P₁₀ = 2.2

P₉₀ = Size of 90(n+1)/100th item

= Size of 90(60+1)/100th item

= Size of 54.9th item

= Size of 54th item + 0.9(size of 55th item - size of 54th item)

= 42 + 0.9(42-42)

P₁₀ = 42

So, we have to calculate the mean of the values which are between 2.2(6th item) & 42(54th item) -

So, 10% trimmed mean is 20.125.

9) -

Variance for the given data is 205.8289.

10) -

Standard deviation for the given data is 14.3667.

11) -

Stem & leaf plot -

We considered stem as tenth place & leaf as unit place.

[Note : For single digits, we take 0 as tenth place digit.]

Stem	Leaf
0	0,0,0,1,2,2,4,5,5,5,7,7,7,7,8,8,8,9
1	0,1,1,2,4,5,5,6,6,6,7,7,8,8,9
2	0,1,2,6,7,8,8,8,9
3	0,1,2,4,5,5,6,8,8
4	0,1,2,2,5,5,6,7,8