3.1 Range quartile deviation mean | unit 3 dispersion and skewness

UNIT – 3

Dispersion and skewness

3.1 Range quartile deviation mean

A range is the most common and easily understandable measure of dispersion. It is the difference between two extreme observations of the data set. If X max and X min are the two extreme observations then

Range = X max – X min

Mean Deviation

Mean deviation is the arithmetic mean of the absolute deviations of the observations from a measure of central tendency. If x1, x2, …, xn are the set of observation, then the mean deviation of x about the average A (mean, median, or mode) is

Mean deviation from average A = 1⁄n [∑|xi – A|]

For a grouped frequency, it is calculated as:

Mean deviation from average A = 1⁄N [∑ fi |xi – A|], N = ∑fi

Quartile Deviation

The quartiles divide a data set into quarters. The first quartile, (Q1) is the middle number between the smallest number and the median of the data. The second quartile, (Q2) is the median of the data set. The third quartile, (Q3) is the middle number between the median and the largest number.

Quartile deviation or semi-inter-quartile deviation is

Q = ½ × (Q3 – Q1)

Quartiles

There are three quartiles, i.e. Q1, Q2 and Q3 which divide the total data into four equal parts when it has been orderly arranged. Q1, Q2 and Q3 are termed as first quartile, second quartile and third quartile or lower quartile, middle quartile and upper quartile, respectively. The first quartile, Q1, separates the first one-fourth of the data from the upper three fourths and is equal to the 25th percentile. The second quartile, Q2, divides the data into two equal parts (like median) and is equal to the 50th percentile. The third quartile, Q3, separates the first three-quarters of the data from the last quarter and is equal to 75th percentile.

Calculation of Quartiles:

The calculation of quartiles is done exactly in the same manner as it is in case of the calculation of median.

The different quartiles can be found using the formula given below:

Qi = l1 + i= 1,2,3

Where,

L1 = lower limit of ith quartile class

L2 = upper limit of ith quartile class

c = cumulative frequency of the class preceding the ith quartile class

f = frequency of ith quartile class.

3.2 Deviation and their coefficients

Range – Range defines the difference between the maximum value and the minimum value given in a data set. More the range , group is more variable. The smaller the range the more homogenous is the group.

R = H – L

Example 1 – 5, 10, 15, 20, 7, 9, 17, 13, 12, 16, 8, 6

Range = H-L

=20 – 5 = 15

Coefficient of range –

Coefficient of range = (15/(20+5))*100 = 60

Example 2 – what is the range for the following set of numbers?

15,21,57,43,11,39,56,83,77,11,64,91,18,37

Solution

Range = H-L

= 91 – 11 = 80

Therefore the range is 80

Example 3 – the frequency table shows the number of goals the lakers scored in their last twenty matches. What was the range

No. of goals	Frequency
0	2
1	3
2	3
3	6
4	3
5	1
6	1
7	1

Solution

The range is the difference between the lowest and highest values.

The highest value was 7 (They scored 7 goals on 1 occasion)

The lowest value was 0 (They scored 0 goals on 2 occasions)

Therefore the range = 7 - 0 = 7

Example 4 – the following table shows the sales of DVD players made by a retail store each month last year

Month	No. of sales
January	25
Feb	43
March	39
April	28
May	29
June	35
July	32
August	46
September	28
October	43
November	51
December	63

Solution

The range is the difference between the lowest and highest values.

The lowest number of sales = 25 in January

The highest number of sales = 63 in December

So the range = 63 - 25 = 38

Example 5 – what is the range for the following set of numbers?

57, -5, 11, 39, 56, 82, -2, 11, 64, 18, 37, 15, 68

Solution

The range is the difference between the lowest and highest values.

The highest value is 82.

The lowest value is -5.

Therefore the range = 82 - (-5) = 82+5 = 87

Merits

Simple and easy to understand

It gives a quick answer

Demerits

It is not based on all observation

Affected by sampling fluctuations

It cannot be calculated in open ended distributions

2. Interquartile range - the interquartile range measures the range of the middle 50% of the values only. It is calculated as the difference between the upper and lower quartile.

Interquartile range = upper quartile – lower quartile

= Q3 – Q1

Examples 1– find the interquartile range for 1, 2, 18, 6, 7, 9, 27, 15, 5, 19, 12.

Solution

Arrange the numbers in ascending order

1, 2, 5, 6, 7, 9, 12, 15, 18, 19, 27

Find the median

Median = 9

(1, 2, 5, 6, 7), 9, (12, 15, 18, 19, 27)

Q1 as median in the lower half and Q3 as median in the upper half

Q1 = median in (1, 2, 5, 6, 7)

Q1 = 5

Q3 = median in (12, 15, 18, 19, 27)

Q3 = 18

Interquartile range = 18 – 5 = 13

Example 2 – find the interquartile for the following data set: 3, 5, 7, 8, 9, 11, 15, 16, 20, 21.

Solution

Arrange the numbers in ascending order

3, 5, 7, 8, 9, 11, 15, 16, 20, 21

Make a mark in the center of the data:

(3, 5, 7, 8, 9,) | (11, 15, 16, 20, 21)

Find the median

Q1 = 7

Q3 = 16

Interquartile range = 16 – 7 = 9

Example 3 - find the interquartile for the following data set: 1, 3, 4, 5, 5, 6, 7, 11

Make a mark in the center of the data:

(1, 3, 4, 5,) (5, 6, 7, 11)

Find the median

Q1 = (3+4)/2 = 3.5

Q3 = (6+7)/2 = 6.5

Interquartile range = 6.5 – 3.5 = 3

Example 4 -

Find the interquartile range for odd sample size

63,64,64,70,72,76,77,81,81

Solution

Make a mark in the center of the data:

(63,64,64,70,)72,(76,77,81,81)

Find the median

Q1 = (64+64)/2 = 64

Q3 = (77+81)/2 = 79

Interquartile range = 79 – 64 = 15

3. Quartile deviation

Quartile deviation is the product of half of the difference between the upper and the lower quartiles.

QD = (Q3 - Q1) / 2

Coefficient of Quartile Deviation = (Q3 – Q1) / (Q3 + Q1)

Quartile deviation for ungrouped data

Examples 1

Day	Frequency
1	20
2	35
3	25
4	12
5	10
6	23
7	18
8	14
9	30
10	40

Solution

Arrange the frequency data in ascending order

Day	Frequency
1	10
2	12
3	14
4	18
5	20
6	23
7	25
8	30
9	35
10	40

First quartile (Q1)

Qi= [i * (n + 1) /4] th observation

Q1= [1 * (10 + 1) /4] th observation

Q1 = 2.75 th observation

Thus, 2.75 th observation lies between the 2nd and 3rd value in the ordered group, between frequency 12 and 14

First quartile (Q1) is calculated as

Q1 = 2nd observation +0.75 * (3rd observation - 2nd observation)

Q1 = 12 + 0.75 * (14 – 12) = 13.50

Third quartile (Q3)

Qi= [i * (n + 1) /4] th observation

Q3= [3 * (10 + 1) /4] th observation

Q3 = 8.25 th observation

So, 8.25 th observation lies between the 8th and 9th value in the ordered group, between frequency 30 and 35

Third quartile (Q3) is calculated as

Q3 = 8th observation +0.25 * (9th observation – 8th observation)

Q3 = 30 + 0.25 * (35 – 30) = 31.25

Now using the quartiles values Q1 and Q3, we will calculate the quartile deviation.

QD = (Q3 - Q1) / 2

QD = (31.25 – 13.50) / 2 = 8.875

Coefficient of Quartile Deviation = (Q3 – Q1) / (Q3 + Q1)

= (31.25 – 13.50) /(31.25 + 13.50) = 0.397

Example 2 – calculate quartile deviation from the following test scores

Sl. N o	Test scores
1	17
2	17
3	26
4	27
5	30
6	30
7	31
8	37

Solution

First quartile (Q1)

Qi= [i * (n + 1) /4] th observation

Q1= [1 * (8 + 1) /4] th observation

Q1 = 2.25 th observation

Thus, 2.25 th observation lies between the 2nd and 3rd value in the ordered group, between frequency 17 and 26

First quartile (Q1) is calculated as

Q1 = 2nd observation +0.75 * (3rd observation - 2nd observation)

Q1 = 17 + 0.75 * (26 – 17) = 23.75

Third quartile (Q3)

Qi= [i * (n + 1) /4] th observation

Q3= [3 * (8 + 1) /4] th observation

Q3 = 6.75 th observation

So, 6.75 th observation lies between the 6th and 7th value in the ordered group, between frequency 30 and 31

Third quartile (Q3) is calculated as

Q3 = 6th observation +0.25 * (7th observation – 6th observation)

Q3 = 30 + 0.25 * (31 – 30) = 30.25

Now using the quartiles values Q1 and Q3, we will calculate the quartile deviation.

QD = (Q3 - Q1) / 2

QD = (30.25 – 23.75) / 2 = 3.25

Quartile deviation for grouped data

Where,

l = lower boundary of quartile group

h = width of quartile group

f = frequency of quartile group

N = total number of observation

C= cumulative frequency preceding quartile group

Example 3

Age in years	40 -44	45 – 49	50 – 54	55 - 59	60 – 64	65 - 69
Employees	5	8	11	10	9	7

Solutions

In the case of Frequency Distribution, Quartiles can be calculated by using the formula:

Class interval	F	Class boundaries	CF
40 -44	5	39.5 – 44.5	5
45 – 49	8	44.5 – 49.5	13
50 – 54	11	49.5 – 54.5	24
55 – 59	10	54.5 – 59.5	34
60 – 64	9	59.5 – 64.5	43
65 – 69	7	64.5 – 69.5	50
Total	50

First quartile (Q1)

Qi= [i * (n ) /4] th observation

Q1 = [1*(50)/4]th observation

Q1 = 12.50th observation

So, 12.50th value is in the interval 44.5 – 49.5

Group of Q1 = 44.5 – 49.5

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q1 = (44.5 + ( 5/ 8)* (1* (50/4) – 5)

Q1 = 49.19

Third quartile (Q3)

Qi= [i * (n) /4] th observation

Q3= [3 * (50) /4] th observation

Q3 = 37.5th observation

So, 37.5th value is in the interval 59.5 – 64.5

Group of Q3 = 59.5 – 64.5

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q3 = (59.5 + ( 5/ 9)* (3* (50/4) – 34)

Q3 = 61.44

QD = (Q3 - Q1) / 2

QD = (61.44 – 49.19) / 2 = 6.13

Coefficient of Quartile Deviation = (Q3 – Q1) / (Q3 + Q1)

= (61.44 – 49.19) /(61.44 + 49.19) = 0.11

Example 4 – computation of quartile deviation for grouped test scores

Class	Frequency
9.3-9.7	22
9.8-10.2	55
10.3-10.7	12
10.8-11.2	17
11.3-11.7	14
11.8-12.2	66
12.3-12.7	33
12.8-13.2	11

Solution

Class	Frequency	Class boundaries	CF
9.3-9.7	2	9.25-9.75	2
9.8-10.2	5	9.75-10.25	2 + 5 = 7
10.3-10.7	12	10.25-10.75	7 + 12 = 19
10.8-11.2	17	10.75-11.25	19 + 17 = 36
11.3-11.7	14	11.25-11.75	36 + 14 = 50
11.8-12.2	6	11.75-12.25	50 + 6 = 56
12.3-12.7	3	12.25-12.75	56 + 3 = 59
12.8-13.2	1	12.75-13.25	59 + 1 = 60

First quartile (Q1)

Qi= [i * (n ) /4] th observation

Q1 = [1*(60)/4]th observation

Q1 = 15th observation

So, 15th value is in the interval 10.25-10.75

Group of Q1 = 10.25-10.75

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q1 = (10.25 + ( 0.5/ 12)* (1* (60/4) – 7)

Q1 = 10.58

Third quartile (Q3)

Qi= [i * (n) /4] th observation

Q3= [3 * (60) /4] th observation

Q3 = 45th observation

So, 45th value is in the interval 11.25-11.75

Group of Q3 = 11.25-11.75

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q3 = (11.25 + ( 0.5/ 14)* (3* (60/4) – 36)

Q3 = 11.57

QD = (Q3 - Q1) / 2

QD = (11.57 – 10.58) / 2 = 0.495

Example 5 – calculate quartile deviation from the following data

CI	F
10 – 15	6
15 – 20	10
20 – 25	15
25 – 30	22
30 – 40	12
40 – 50	9
50 – 60	4
60 - 70	2

Solution

CI	F	Cf
10 – 15	6	6
15 – 20	10	16
20 – 25	15	31
25 – 30	22	53
30 – 35	12	65
35 – 40	9	74
45 – 50	4	78
55 – 60	2	80

First quartile (Q1)

Qi= [i * (n ) /4] th observation

Q1 = [1*(80)/4]th observation

Q1 = 20th observation

So, 20th value is in the interval 20 - 25

Group of Q1 = 20 - 25

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q1 = (20 + ( 5/ 15)* (1* (80/4) – 16)

Q1 = 21.33

Third quartile (Q3)

Qi= [i * (n) /4] th observation

Q3= [3 * (80) /4] th observation

Q3 = 60th observation

So, 60th value is in the interval 30 - 35

Group of Q3 = 30 - 35

Qi = (I + (h / f) * ( i * (N/4) – c) ; i = 1,2,3

Q3 = (30 + ( 5/ 12)* (3* (80/4) – 53)

Q3 = 32.91

QD = (Q3 - Q1) / 2

QD = (32.91 – 21.33) / 2 = 5.79

Merits

It provide better result than range mode

It is not effected by extreme values

Demerits

It is completely dependent on central item

All items are not taken onto consideration

4. Mean deviation – The average of the absolute values of deviation from the mean, median or mode is called mean deviation. This method removes shortcoming of range and QD.

Where, ∑ is total of;

X is the score, X is the mean, and N is the number of scores

D = Deviation of individual scores from mean

Example 1 –

Computation of mean deviation in ungrouped data

X = 55, 45, 39, 41, 40, 48, 42, 53, 41, 56

Solution

X	(X – X )	Absolute deviation (signed ignored)
55	55 - 46 = 9	9
45	45 – 46 = -1	1
39	-7	7
41	-5	5
40	-6	6
48	2	2
42	-4	4
53	7	7
41	-5	5
56	10	10
∑X = 460		∑ X – X = 56

Mean = 460/10 = 46

MD = 56/10 = 5.6

Example 2- Peter did a survey on the number of pets owned by his classmates, with the following result. What is the mean deviation of the number of pets?

No. of pets	Frequency
0	4
1	12
2	8
3	2
4	1
5	2
6	1

Solution

X	F	Fx	(X – X )	f(X – X )
0	4	0	1.8	7.2
1	12	12	0.8	9.6
2	8	16	0.2	1.6
3	2	6	1.2	2.4
4	1	4	2.2	2.2
5	2	10	3.2	6.4
6	1	6	3.2	4.2
	30	54	4.2	33.6

Mean = 54/30 = 1.8

MD = 33.6/30 = 1.12

Computation of Mean deviation in grouped data

Example 3 -

Class interval	15 – 19	20 – 24	25 – 29	30 – 34	35 – 39	40 – 44	45 - 49
Frequency	1	4	6	9	5	3	2

Class Interval	F	X	FX	D	FD
15 – 19	1	17	17	15	15
20 – 24	4	22	88	10	40
25 – 29	6	27	162	5	30
30 - 34	9	32	288	0	0
35 - 39	5	37	185	5	25
40 - 44	3	42	126	10	30
45 - 49	2	47	94	15	30
	N = 30		∑fx = 960		= 170

Mean = 960/30 = 32

MD = 170 / 30 = 5.667

Coefficient of mean deviation

Coefficient of mean deviation = (5.67/32)*100 = 17.71

Example 4 – Calculate mean deviation from the median.

Class	5 -15	15 - 25	25 - 35	35 - 45	45 – 55
Frequency	5	9	7	3	8

Solution

x	f	cf	Midpoint x	x –median	F(x-m)
5 -15	5	5	10	17.42	87.1
15 -25	9	14	20	7.42	66.78
25 -35	7	21	30	2.58	18.06
35 -45	3	24	40	12.58	37.74
45- 55	8	32	50	22.58	180.64
	32				390.32

Since n/2 = 32/2 = 16, therefore the class is 25 – 35 is the median.

Median =

Median = 25+16-14 *10 = 27.42

MD from median is 390.32/32 = 12.91

Example 5 – calculate the mean deviation from continuous frequency distribution

Age group	15 - 25	25 - 35	35 - 45	45 - 55
No. of people	25	54	34	20

Solution

Age group (X)	Number of people (f)	Midpoint x	fx	X - x	f( x – x )
15 – 25	25	20	500	13.684	324.1
25 – 35	54	30	1620	3.684	198.936
35 – 45	34	40	1360	6.316	214.744
45 - 55	20	50	1000	16.316	352.32
	133				1090.1

Mean = 4480/133 = 33.684

MD = 1090.1/133 = 8.196

Merits

It is easy to calculate

It helps in making comparison

It is not affected by extreme items

Demerits

It ignores algebraic sign. And are not used for mathematical treatment

It is not reliable

5. Standard deviation – standard deviation is calculated as square root of average of squared deviations taken from actual mean. It is also called root mean square deviation. This measure suffers from less drawbacks and provides accurate results. It removes the drawbacks of ignoring algebraic sign. We square the deviation to make them positive.

Two ways of computing SD

Direct method

2. Shortcut method

d = Deviation of the score from an assumed mean, say AM; i.e. d = (X – AM). AM is assumed mean

d2 = the square of the deviation.

∑d = the sum of the deviations.

∑d2 = the sum of the squared deviations.

N = No. of the scores

Standard deviation in ungrouped data

Direct method

Example 1–

X = 12, 15, 10, 8, 11, 13, 18, 10, 14, 9

Mean = 120/10 = 12

Scores	d
12	12-12 = 0	0
15	15-12 = 3	9
10	10 -12 = -2	4
8	-4	16
11	-1	1
13	1	1
18	6	36
10	-2	4
14	2	4
9	-3	9
= 120	= 0	= 84

= 2.9

2. Shortcut method

Assumed mean (AM) = 11

Scores	D = (X- AM)
12	12-11 = 1	1
15	15-11 = 4	16
10	10 -11 = -1	1
8	-3	9
11	0	0
13	2	4
18	7	49
10	--1	1
14	3	9
9	-2	4
= 120	= 10	= 94

SD from short cut method = 2.9

Example 2 –Ram did a survey of the number of pets owned by his classmates, with the following results

No. of pets	Frequency
0	4
1	12
2	8
3	2
4	1
5	2
6	1

Solution

x	f	fx	X – x	(x – x ) 2	F (x – x ) 2
0	4	0	-1.8	3.24	12.96
1	12	12	-0.8	0.64	7.68
2	8	16	0.2	0.04	0.32
3	2	6	1.2	1.44	2.88
4	1	4	2.2	4.84	4.84
5	2	10	3.2	10.24	20.48
6	1	6	4.2	17.64	17.64
	30	54			66.80

Mean = 54/30 = 1.8

SD = √66.80/30 = 1.49

Standard deviation in grouped data

Direct method

Example 3 –

C.I.	0 - 2	3 - 5	6- 8	9-11	12-14	15 -17	18 - 20
F	1	3	5	7	6	5	3

Solution

C.I	f	Midpoint x	Fx	d		fd2
0-2	1	1	1	-10.1	102.01	102.01
3-5	3	4	12	-7.1	50.41	151.23
6-8	5	7	35	-4.1	16.81	84.05
9-11	7	10	70	-1.1	1.21	8.47
12-14	6	13	78	1.9	3.61	21.66
15-17	5	16	80	4.9	24.01	120.05
18-20	3	19	57	7.9	62.41	187.23
	30		333			674.70

Mean = 333/30 = 11.1

SD =

Shortcut method

C.I	f	Midpoint x	d(X-AM)	fd	fd2
0-2	1	1	-9	-9	81
3-5	3	4	-6	-18	108
6-8	5	7	-3	-15	45
9-11	7	10	0	0	0
12-14	6	13	3	18	54
15-17	5	16	6	30	180
18-20	3	19	9	27	243
	30			33	711

Assumed mean = 10

Step deviation method

C.I	f	Midpoint x	d	fd	fd2
0-2	1	1	-3	-3	9
3-5	3	4	-2	-6	12
6-8	5	7	-1	-5	5
9-11	7	10	0	0	0
12-14	6	13	1	6	6
15-17	5	16	2	10	20
18-20	3	19	3	9	27
	30			11	79

Here, d is calculate as (X –AM)/i, where i is length of class interval

d = (1 -10)/3 = -3 and so on

Coefficient of standard deviation

Coefficient of SD = (4.74/11.1)*100 = 42.70

Example 4 – calculate the standard deviation using the direct method

Class interval	Frequency
30 – 39	3
40 – 49	1
50 – 59	8
60 – 69	10
70 – 79	7
80 – 89	7
90 – 99	4

Solution

Class interval	Frequency	Midpoint x	fx	X – x	(x – x ) 2	F (x – x ) 2
30 – 39	3	34.5	103.5	-33.5	1122.25	3366.75
40 – 49	1	44.5	44.5	-23.5	552.25	552.25
50 – 59	8	54.5	436.0	-13.5	182.25	1458
60 – 69	10	64.5	645.0	-3.5	12.25	122.5
70 – 79	7	74.5	521.5	6.5	42.25	295.75
80 – 89	7	84.5	591.5	16.5	272.25	1905.75
90 – 99	4	94.5	378.0	26.5	702.25	2809
	40		2720			10510

Mean = 2720/40 = 68

SD = √10510/40 = 16.20

Example 5 - calculate the mean and standard deviation of hours spent watching television by the 220 students.

Hours	No. of students
10 – 14	2
15 – 19	12
20 – 24	23
25 – 29	60
30 – 34	77
35 – 39	38
40 - 44	8

Solution

Hours	No. of students	x	fx	X – x	(x – x ) 2	F (x – x ) 2
10 – 14	2	12	24	-17.82	317.49	634.98
15 – 19	12	17	204	-12.82	164.31	1971.67
20 – 24	23	22	506	-7.82	61.12	1405.85
25 – 29	60	27	1620	-2.82	7.94	476.53
30 – 34	77	32	2464	2.18	4.76	366.55
35 – 39	38	37	1406	7.18	51.58	1959.98
40 - 44	8	42	336	12.18	148.40	1187.17
	220		6560			8002.73

Mean = 6560/220 = 29.82

SD = √8002.73/220 = 6.03

Merits

It takes into account all the items and are used for future statistical analysis

It is suitable for making comparison

Demerits

It is difficult to compute

Key takeaways:

Dispersion measures the scatteredness of the data series around its average.

Coefficient of skewness

What is Skewness?

Skewness is the measure of the asymmetry of an ideally symmetric probability distribution and is given by the third standardized moment. If that sounds way too complex, don’t worry! Let me break it down for you.

In simple words, skewness is the measure of how much the probability distribution of a random variable deviates from the normal distribution. Now, you might be thinking – why am I talking about normal distribution here?

Well, the normal distribution is the probability distribution without any skewness. You can look at the image below which shows symmetrical distribution that’s basically a normal distribution and you can see that it is symmetrical on both sides of the dashed line. Apart from this, there are two types of skewness:

Positive Skewness

Negative Skewness

Coefficient of Skewness:

Skewness is a measure of symmetry or lack of symmetry in a distribution. A distribution is symmetric if it looks same both its left and right side. The skewness for normal distribution is Zero. Negative values for skewness indicate that the data are skewed left and positive values for skewness indicate that the data are skewed right. For small data sets, this measure is unreliable. The below diagram shows how a normal distribution curve looks like in different situation of skewness:

The formula for measuring Coefficient of Skewness as given by Karl Pearson is as under:

Where, = Karl Pearson’s Coefficient of Skewness

= Standard Deviation

The formula for measuring Co-efficient of Skewness as given by Bowley is as under:

Example 1: From the following data, calculate Karl Pearson’s Co-efficient of Skewness:

Mean = 16, Mode = 38, Standard Deviation = 5

Solution: We know that

= -4.4

Example 2 – Below are the data of hours spent watching television by the 220 students. Calculate Karl Pearsons Co-efficient of Skewness.

Hours	No. of students
10 – 14	2
15 – 19	12
20 – 24	23
25 – 29	60
30 – 34	77
35 – 39	38
40 - 44	8

Solution:

Hours	No. of students	x	fx	X – x	(x – x ) 2	F (x – x ) 2
10 – 14	2	12	24	-17.82	317.49	634.98
15 – 19	12	17	204	-12.82	164.31	1971.67
20 – 24	23	22	506	-7.82	61.12	1405.85
25 – 29	60	27	1620	-2.82	7.94	476.53
30 – 34	77	32	2464	2.18	4.76	366.55
35 – 39	38	37	1406	7.18	51.58	1959.98
40 - 44	8	42	336	12.18	148.40	1187.17
	220		6560			8002.73

Mean = 6560/220 = 29.82

SD = √8002.73/220 = 6.03

Mode = L1 + (L2 – L1) d1

d1 +d2

Here modal class is 30 – 34 (Since the frequency is highest)

L1 = 30, L2 = 34, d1 = 17, d2 = 39

Mode = 30 + (34 – 30) 17

17 + 39

Mode = 30 + x 17

= 30 + 1.21

= 31.21

Therefore, Co-efficient of Skewness

= - 0.23

Example 3: Calculate Bowley’s Coefficient of Skewness from the following test scores:

Sl. N o	Test scores
1	17
2	17
3	26
4	27
5	30
6	30
7	31
8	37

Solution:

First quartile (Q1)

Qi= [i * (n + 1) /4] th observation

Q1= [1 * (8 + 1) /4] th observation

Q1 = 2.25 th observation

Thus, 2.25 th observation lies between the 2nd and 3rd value in the ordered group, between frequency 17 and 26

First quartile (Q1) is calculated as

Q1 = 2nd observation +0.75 * (3rd observation - 2nd observation)

Q1 = 17 + 0.75 * (26 – 17) = 23.75

Second quartile()

Q2= [2 * (8 + 1) /4] th observation

Q2 = 4.5th Observation

So, 4.5th observation lies between 4th and 5th value in ordered group, between frequency 27 and 30.

Hence Q2 = 4th observation + 0.50 * (5th observation – 6th observation)

Q2 = 27 + 0.50 * (30 – 27) = 28.5

Third quartile (Q3)

Qi= [i * (n + 1) /4] th observation

Q3= [3 * (8 + 1) /4] th observation

Q3 = 6.75 th observation

So, 6.75 th observation lies between the 6th and 7th value in the ordered group, between frequency 30 and 31

Third quartile (Q3) is calculated as

Q3 = 6th observation +0.25 * (7th observation – 6th observation)

Q3 = 30 + 0.25 * (31 – 30) = 30.25

Therefore, Bowley’s Coefficient of Skewness is calculated as under:

= = - 0.461

Key takeaways –

Skewness is the measure of the asymmetry of an ideally symmetric probability distribution and is given by the third standardized moment.

Moments

Moments are a set of statistical parameters to measure a distribution. Four moments are commonly used:

• 1st moment - Mean (describes central value)

• 2nd moment - Variance (describes dispersion)

• 3rd moment - Skewness (describes asymmetry)

• 4th moment - Kurtosis (describes peakedness)

The formula for calculating moments is as follows:

1st moment =

2nd moment =

3rd moment =

4th moment =

Raw Moments and Central Moments

The n-th moment about zero of a probability density function f(x) is the expected value of Xn and is called a raw moment or crude moment. The moments about its mean μ are called central moments; these describe the shape of the function, independently of translation.

A moment mu_n of a probability function P(x) taken about 0,

			(1)
			(2)

The raw moments mu_n^' (sometimes also called "crude moments") can be expressed as terms of the central moments mu_n (i.e., those taken about the mean ) using the inverse binomial transform

mu_n^'=sum_(k=0)^n(n; k)mu_kmu_1^('n-k),

(3)

with mu_0=1 and mu_1=0 (Papoulis 1984, p. 146). The first few values are therefore

			(4)
			(5)
			(6)
			(7)

The raw moments mu_n^' can also be expressed in terms of the cumulates kappa_n by exponentiating both sides of the series

lnphi=ln(sum_(k=0)^infty((it)^k)/(k!)mu_k^')=sum_(n=0)^inftykappa_n((it)^n)/(n!),

(8)

where phi is the characteristic function, to obtain

sum_(k=0)^infty((it)^k)/(k!)mu_k^'=exp(sum_(n=0)^inftykappa_n((it)^n)/(n!)).

(9)

The first few terms are then given by

			(10)
			(11)
			(12)
			(13)
			(14)

The raw moment of a multivariate probability function P(x_1,x_2,...) can be similarly defined as

mu_(m,n,...)^'=<x_1^mx_2^n...>.

(15)

Therefore,

mu_(n,0,...,0)^'=mu_n^'.

(16)

The multivariate raw moments can be expressed in terms of the multivariate cumulants. For example,

			(17)
			(18)

In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location.

Key Takeaways:

Moments are a set of statistical parameters to measure a distribution.

The n-th moment about zero of a probability density function f(x) is the expected value of Xn and is called a raw moment or crude moment. The moments about its mean μ are called central moments.

3.3 Standards deviation coefficient of variation skewness and its coefficients.

Standard deviation is calculated as square root of average of squared deviations taken from actual mean. It is also called root mean square deviation. This measure suffers from less drawbacks and provides accurate results. It removes the drawbacks of ignoring algebraic sign. We square the deviation to make them positive.

Two ways of computing SD

3. Direct method

4. Shortcut method

d = Deviation of the score from an assumed mean, say AM; i.e. d = (X – AM). AM is assumed mean

d2 = the square of the deviation.

∑d = the sum of the deviations.

∑d2 = the sum of the squared deviations.

N = No. of the scores

Standard deviation in ungrouped data

3. Direct method

Example 1–

X = 12, 15, 10, 8, 11, 13, 18, 10, 14, 9

Mean = 120/10 = 12

Scores	d
12	12-12 = 0	0
15	15-12 = 3	9
10	10 -12 = -2	4
8	-4	16
11	-1	1
13	1	1
18	6	36
10	-2	4
14	2	4
9	-3	9
= 120	= 0	= 84

= 2.9

4. Shortcut method

Assumed mean (AM) = 11

Scores	D = (X- AM)
12	12-11 = 1	1
15	15-11 = 4	16
10	10 -11 = -1	1
8	-3	9
11	0	0
13	2	4
18	7	49
10	--1	1
14	3	9
9	-2	4
= 120	= 10	= 94

SD from short cut method = 2.9

Example 2 –Ram did a survey of the number of pets owned by his classmates, with the following results

No. of pets	Frequency
0	4
1	12
2	8
3	2
4	1
5	2
6	1

Solution:

x	f	fx	X – x	(x – x ) 2	F (x – x ) 2
0	4	0	-1.8	3.24	12.96
1	12	12	-0.8	0.64	7.68
2	8	16	0.2	0.04	0.32
3	2	6	1.2	1.44	2.88
4	1	4	2.2	4.84	4.84
5	2	10	3.2	10.24	20.48
6	1	6	4.2	17.64	17.64
	30	54			66.80

Mean = 54/30 = 1.8

SD = √66.80/30 = 1.49

Standard deviation in grouped data

Direct method

Example 3 –

C.I.	0 - 2	3 - 5	6- 8	9-11	12-14	15 -17	18 - 20
F	1	3	5	7	6	5	3

Solution:

C.I	f	Mid-point x	Fx	d		fd2
0-2	1	1	1	-10.1	102.01	102.01
3-5	3	4	12	-7.1	50.41	151.23
6-8	5	7	35	-4.1	16.81	84.05
9-11	7	10	70	-1.1	1.21	8.47
12-14	6	13	78	1.9	3.61	21.66
15-17	5	16	80	4.9	24.01	120.05
18-20	3	19	57	7.9	62.41	187.23
	30		333			674.70

Mean = 333/30 = 11.1

SD =

Shortcut method

C.I	f	Mid-point x	d(X-AM)	fd	fd2
0-2	1	1	-9	-9	81
3-5	3	4	-6	-18	108
6-8	5	7	-3	-15	45
9-11	7	10	0	0	0
12-14	6	13	3	18	54
15-17	5	16	6	30	180
18-20	3	19	9	27	243
	30			33	711

Assumed mean = 10

Step deviation method

C.I	f	Mid-point x	d	fd	fd2
0-2	1	1	-3	-3	9
3-5	3	4	-2	-6	12
6-8	5	7	-1	-5	5
9-11	7	10	0	0	0
12-14	6	13	1	6	6
15-17	5	16	2	10	20
18-20	3	19	3	9	27
	30			11	79

Here, d is calculated as (X –AM)/i, where i is length of class interval

d = (1 -10)/3 = -3 and so on

Coefficient of standard deviation

Coefficient of SD = (4.74/11.1)*100 = 42.70

Example 4 – calculate the standard deviation using the direct method

Class interval	Frequency
30 – 39	3
40 – 49	1
50 – 59	8
60 – 69	10
70 – 79	7
80 – 89	7
90 – 99	4

Solution:

Class interval	Frequency	Mid-point x	fx	X – x	(x – x ) 2	F (x – x ) 2
30 – 39	3	34.5	103.5	-33.5	1122.25	3366.75
40 – 49	1	44.5	44.5	-23.5	552.25	552.25
50 – 59	8	54.5	436.0	-13.5	182.25	1458
60 – 69	10	64.5	645.0	-3.5	12.25	122.5
70 – 79	7	74.5	521.5	6.5	42.25	295.75
80 – 89	7	84.5	591.5	16.5	272.25	1905.75
90 – 99	4	94.5	378.0	26.5	702.25	2809
	40		2720			10510

Mean = 2720/40 = 68

SD = √10510/40 = 16.20

Example 5 - calculate the mean and standard deviation of hours spent watching television by the 220 students.

Hours	No. of students
10 – 14	2
15 – 19	12
20 – 24	23
25 – 29	60
30 – 34	77
35 – 39	38
40 - 44	8

Solution:

Hours	No. of students	x	fx	X – x	(x – x ) 2	F (x – x ) 2
10 – 14	2	12	24	-17.82	317.49	634.98
15 – 19	12	17	204	-12.82	164.31	1971.67
20 – 24	23	22	506	-7.82	61.12	1405.85
25 – 29	60	27	1620	-2.82	7.94	476.53
30 – 34	77	32	2464	2.18	4.76	366.55
35 – 39	38	37	1406	7.18	51.58	1959.98
40 - 44	8	42	336	12.18	148.40	1187.17
	220		6560			8002.73

Mean = 6560/220 = 29.82

SD = √8002.73/220 = 6.03

Merits

It takes into account all the items and are used for future statistical analysis

It is suitable for making comparison

Demerits

It is difficult to compute

3.6 Coefficient of Variation

Standard Variation is an absolute measure of dispersion. When comparison between two series has to be made, coefficient of variation is used. Coefficient of variation is a statistical measure of the dispersion of data with respect to mean.

Where, σ = Standard Deviation

µ = Mean

Example 1: Calculate Coefficient of Variation from the following data given below:

Standard Deviation = 4.5 Mean= 12.

Solution: We know,

Coefficient of Variation = x 100%

= x 100%

= 37.5%

Example 2: Calculate Standard Deviation and Co-efficient of Variation.

Class interval	Frequency
30 – 39	3
40 – 49	1
50 – 59	8
60 – 69	10
70 – 79	7
80 – 89	7
90 – 99	4

Solution:

Class interval	Frequency	Mid-point x	fx	X – x	(x – x ) 2	F (x – x ) 2
30 – 39	3	34.5	103.5	-33.5	1122.25	3366.75
40 – 49	1	44.5	44.5	-23.5	552.25	552.25
50 – 59	8	54.5	436.0	-13.5	182.25	1458
60 – 69	10	64.5	645.0	-3.5	12.25	122.5
70 – 79	7	74.5	521.5	6.5	42.25	295.75
80 – 89	7	84.5	591.5	16.5	272.25	1905.75
90 – 99	4	94.5	378.0	26.5	702.25	2809
	40		2720			10510

Mean = 2720/40 = 68

SD = √10510/40 = 16.20

Coefficient of Variation = x 100%

= x 100%

= 23.82 %

Coefficient of Skewness:

The formula for measuring Coefficient of Skewness as given by Karl Pearson is as under:

Where, = Karl Pearson’s Coefficient of Skewness

= Standard Deviation

The formula for measuring Co-efficient of Skewness as given by Bowley is as under:

Example 1: From the following data, calculate Karl Pearson’s Co-efficient of Skewness:

Mean = 16, Mode = 38, Standard Deviation = 5

Solution: We know that

= -4.4

Example 2 – Below are the data of hours spent watching television by the 220 students. Calculate Karl Pearson’s Co-efficient of Skewness.

Hours	No. of students
10 – 14	2
15 – 19	12
20 – 24	23
25 – 29	60
30 – 34	77
35 – 39	38
40 - 44	8

Solution:

Hours	No. of students	x	fx	X – x	(x – x ) 2	F (x – x ) 2
10 – 14	2	12	24	-17.82	317.49	634.98
15 – 19	12	17	204	-12.82	164.31	1971.67
20 – 24	23	22	506	-7.82	61.12	1405.85
25 – 29	60	27	1620	-2.82	7.94	476.53
30 – 34	77	32	2464	2.18	4.76	366.55
35 – 39	38	37	1406	7.18	51.58	1959.98
40 - 44	8	42	336	12.18	148.40	1187.17
	220		6560			8002.73