statistics measures of central tendency (averages
TRANSCRIPT
15.1
STATISTICS โ MEASURES OF CENTRAL TENDENCY (AVERAGES) &
DISPERSION
DEFINITION OF CENTRAL TENDENCY / AVERAGES:
Central tendency (tending to the central value), which helps for finding performance and comparison
X ๐
00-19 1 Minimum
20-39 3 Gradually increasing
40-59 7 Maximum
60-79 2 Gradually decreasing
80-99 1 Minimum
X - (Any variable: Height, Weight, Marks, Profits, Wages, and so on)
๐ - Frequency, (Usually, repetitiveness, frequent happenings, number of times of occurrence)
List of Formula
Arithmetic Mean (๏ฟฝฬ ๏ฟฝ) Geometric Mean (๐ฎ๐ด) Harmonic Mean (๐ฏ๐ด)
Weighted Average
Xฬ =โ ๐ค๐
โ ๐ค G = (๐1
๐ค1 ร ๐2๐ค2 ร โฆ ร ๐๐
๐ค๐)1
โ ๐ค
Or ๐บ = ๐ด๐๐ก๐๐๐๐ (โ ๐ค ๐๐๐๐
โ ๐ค)
๐ป =โ ๐ค
โ๐ค
๐
Combined Mean
xฬ =๐1xฬ 1 + ๐2xฬ 2
๐1 + ๐2
๐บ = ๐ด๐๐ก๐๐๐๐ (๐1 log ๐บ1 + ๐2 log ๐บ2
๐1 + ๐2
) ๐ป =๐1 + ๐2๐1
๐ป1+
๐2
๐ป2
15.2
Measures of Central Tendency (Averages)
Mean Partition Values: (Arrange the items in ascending order)
Mode (๐ด๐) Arithmetic
(usual cases)
(Direct Method)
Geometric
(Comparisons
โ ratios,
Proportions and %)
Harmonic
(Two units together
E.g. speed =
distance / time
Median (๐ด๐) Fractiles (๐ญ๐)
Individual
Xฬ =๐1 + ๐2 โฆ ๐๐
๐
Xฬ =โ ๐๐
๐๐=1
๐
xฬ =โ ๐
๐
GM = (๐1. ๐2. โฆ ๐๐)1
๐
๐๐ ๐บ๐ = ๐ด๐๐ก๐๐๐๐ (โ ๐๐๐ ๐
๐)
๐ป๐ =๐
โ1
๐
If โnโ is odd:
๐๐ = (๐ + 1
2)
๐กโ
๐๐๐
(i.e. the middle obs)
If โnโ is even:
๐๐ =(
๐
2)
๐กโ+ (
๐
2+ 1)
๐กโ ๐๐๐
2
๐น๐ =๐(๐ + 1)
๐น
๐๐ = ๐๐๐ ๐ก ๐ข๐ ๐ข๐๐
(๐๐๐๐๐ก๐๐๐ ๐๐๐กโ๐๐)
Discrete series
Xฬ =โ ๐๐
โ ๐ =
โ ๐๐
๐
= (๐1๐1 . ๐2
๐2 . โฆ ๐๐๐๐)
1
๐
๐๐ ๐บ = ๐ด๐๐ก๐๐๐๐ (โ ๐ ๐๐๐ ๐
๐)
๐ป๐ =๐
โ๐
๐
๐๐ = ๐ ๐๐ง๐ ๐๐ (๐ถ๐ >๐ + 1
2) ๐น๐ = ๐ ๐๐ง๐ ๐๐ (๐ถ๐ >
๐(๐ + 1)
๐น)
Regular frequency
๐๐ = ๐๐๐๐๐ก๐๐๐ ๐๐๐กโ๐๐
Irregular frequency
๐๐ = ๐๐๐๐ข๐๐๐๐ ๐๐๐กโ๐๐
Continuous / Grouped Frequency / (Interpolation Method)
Xฬ =โ ๐๐
โ ๐ =
โ ๐๐
๐
G = (๐1๐1 . ๐2
๐2 . โฆ ๐๐๐๐)
1
๐
๐ถ๐ ๐บ = ๐ด๐๐ก๐๐๐๐ (โ ๐ ๐๐๐ ๐
๐)
๐ป๐ =๐
โ๐
๐
๐๐ = ๐1 + (
๐
2โ ๐๐
๐๐ข โ ๐๐) ร ๐ถ
๐ถ๐ ๐ +
๐
2โ ๐
๐ร ๐
๐น๐ = ๐1 + (๐
๐
๐นโ ๐๐
๐๐ข โ ๐๐) ร ๐ถ
๐ถ๐ ๐ +๐
๐
๐นโ ๐
๐ร ๐
๐๐ = ๐1 + (๐0 โ ๐โ1
2๐0 โ ๐โ1 โ ๐1) ร ๐ถ
Note:
1. Indirect / Shortcut / Assumed Mean (A) Method: Deviation Method (๐ = ๐ โ ๐ด): Xฬ = ๐ด +โ ๐
๐ & Step-Deviation Method (๐ =
๐โ๐ด
๐ถ): xฬ = ๐ด +
โ ๐
๐ร ๐ถ
2. Empirical relationship (thumb rule): If mode is ill-defined (๐๐ ๐๐๐ ๐ ๐๐ ๐๐๐๐๐๐๐ก๐๐๐ฆ ๐ ๐๐๐ค๐๐ ๐๐๐ ๐ก๐๐๐๐ข๐ก๐๐๐): Xฬ โ ๐๐ = 3(Xฬ โ ๐๐) ๐๐ ๐๐ = 3๐๐ โ 2Xฬ
3. Fractiles: Quartiles (Q), Octiles (O), Deciles (D) and Percentiles (P)
15.3
Measures of Dispersion
Absolute Relative
(i) ๐๐๐ง๐ ๐ (๐) = ๐ฟ โ ๐ ๐๐จ๐๐๐๐ข๐๐ข๐๐ง๐ญ ๐จ๐ ๐ซ๐๐ง๐ ๐(๐ถ๐ ๐ ) =
๐ฟ โ ๐
๐ฟ + ๐ร 100
(ii) ๐๐ฎ๐๐ซ๐ญ๐ข๐ฅ๐ ๐๐๐ฏ๐ข๐๐ญ๐ข๐จ๐ง (๐๐) =
๐3 โ ๐1
2
(Otherwise Semi inter quartile range)
๐๐ง๐ญ๐๐ซ ๐ช๐ฎ๐๐ซ๐ญ๐ข๐ฅ๐ ๐ซ๐๐ง๐ ๐ = ๐3 โ ๐1
Coefficient of Quartile Deviation (Co QD)
๐ถ๐ ๐๐ท =๐3 โ ๐1
๐3 + ๐1
ร 100
(iii) Mean Deviation (MD) about A, (๐จ = Xฬ , ๐๐ , ๐๐) Coefficient of Mean Deviation (๐ถ๐ ๐๐ท๐ด)
๐๐ง๐๐ข๐ฏ๐ข๐๐ฎ๐๐ฅ: M๐ท๐ด =
1
๐โ|๐ฅ โ ๐ด| ๐ถ๐ ๐๐ท๐ด =
๐๐ท๐ด
๐ดร 100
๐๐ข๐ฌ๐๐ซ๐๐ญ๐: M๐ท๐ด =
1
๐โ ๐|๐ฅ โ ๐ด|
๐๐จ๐ง๐ญ๐ข๐ง๐ฎ๐จ๐ฎ๐ฌ: ๐๐ท๐ด =
1
๐โ ๐|๐ โ ๐ด|
(iv) Standard Deviation (s) Coefficient of Variation (CV)
๐๐ง๐๐ข๐ฏ๐ข๐๐ฎ๐๐ฅ: ๐ = โโ(๐ โ Xฬ )2
๐ ๐๐โ
โ ๐2
๐โ Xฬ 2
๐ถ๐ =๐
Xฬ ร 100
๐๐ข๐ฌ๐๐ซ๐๐ญ๐: ๐ = โโ ๐(๐ โ Xฬ )2
๐ ๐๐โ
โ ๐๐2
๐โ Xฬ 2
๐๐จ๐ง๐ญ๐ข๐ง๐ฎ๐จ๐ฎ๐ฌ: ๐ = โโ ๐(๐ โ Xฬ )2
๐ ๐๐ โ
โ ๐๐2
๐โ Xฬ 2
๐๐๐๐๐๐๐๐ = ๐ 2
Shortcut:
๐ = โโ ๐๐2
๐โ (
โ ๐๐
๐)
2
๐โ๐๐๐ ๐ = ๐ โ ๐ด (๐๐๐ ๐๐๐๐๐ฃ๐๐๐ข๐๐ ๐๐๐ ๐๐๐ ๐๐๐๐ก๐) & ๐ =๐ โ ๐ด
๐ถ ๐๐๐ ๐๐๐๐ก๐๐๐ข๐๐ข๐
Comparison
Absolute Measure Relative Measure
1 Dependent of unit Independent of unit
2 Not considered for comparison considered for comparison
3 Not much difficult compared to Relative measure Difficult to compute and comprehend.
15.4
INDIVIDUAL OBSERVATIONS
Question 1: From the Individual Observations: 3, 6, 48 & 24, find out the following
Measures of Averages Measures of Dispersion
Arithmetic Mean Absolute Measure Relative Measure
Geometric Mean Range Coefficient of Range
Harmonic Mean Quartile Deviation Coefficient of Quartile Deviation
Median Mean Deviation Coefficient of Mean Deviation
Fractiles (๐1, ๐3, ๐6, ๐ท7 & ๐75) Standard Deviation / Variation Coefficient of Variation
Mode
Answer:
Measures of Averages
Mean Formula Calculation Answer
AM Xฬ =
โ ๐
๐
3 + 6 + 24 + 48
4
81
4
20.25
GM GM = (๐1 ร ๐2 ร โฆ ร ๐)1
๐ (3 ร 6 ร 24 ร 48)1
4 (34. 44)1
4 12
HM ๐ป๐ =๐
โ1
๐
4
1
3+
1
6+
1
24+
1
48
4 ร 48
16 + 8 + 2 + 1=
192
27
7.11
Note:
๐ฟ 3 6 24 48
๐ฅ๐จ๐ ๐ฟ 0.4771 0.7782 1.3802 1.6812
โ log ๐ 4.3167
Formula Calculation Answer
GM ๐ด๐๐ก๐๐๐๐ (
โ log ๐
๐) ๐ด๐๐ก๐๐๐๐ (
4.3167
4)
11.94
Positional Average
Formula Calculations Answer
๐๐ ๐ ๐๐ง๐ ๐๐ (๐ + 1
2)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 2.5๐กโ ๐๐๐
6 + 0.5(24 โ 6) 15 2๐๐ ๐๐๐ + 0.5 (3๐๐ ๐๐๐ โ 2๐๐ ๐๐๐ )
๐1 ๐ ๐๐ง๐ ๐๐ (๐ + 1
2)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 1.25๐กโ ๐๐๐
3 + 0.25(6 โ 3) 3.75 1๐ ๐ก ๐๐๐ + 0.25 (2๐๐ ๐๐๐ โ 1๐ ๐ก ๐๐๐ )
๐3 ๐ ๐๐ง๐ ๐๐ (3(๐ + 1)
4)
๐กโ
๐๐๐
๐๐๐ง๐ ๐๐ 3.75๐กโ ๐๐๐
3๐๐ ๐๐๐ + 0.75 (4๐กโ ๐๐๐ โ 3๐๐ ๐๐๐ )
24 + 0.75 (48 โ 24)
๐๐
๐75 ๐ ๐๐ง๐ ๐๐ (75(๐ + 1)
100)
๐กโ
๐๐๐
๐6 ๐ ๐๐ง๐ ๐๐ (6(๐ + 1)
8)
๐กโ
๐๐๐
๐ต๐๐๐: ๐3 = ๐6 = ๐75
15.5
๐ท7 ๐ ๐๐ง๐ ๐๐ (7(๐ + 1)
10)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 3.5๐กโ ๐๐๐
24 + 0.5(48 โ 24) ๐๐ 3๐๐ ๐๐๐ + 0.5 (4๐กโ ๐๐๐ โ 3๐๐ ๐๐๐ )
Mode
Mode is ill-defined (Since all the observation has equal appearance)
Hence, the empirical relation is used to arrive ๐๐
Formula Calculations Answer
๐๐ ๐๐๐๐ โ ๐๐๐๐ = 3(๐๐๐๐ โ ๐๐๐๐๐๐) 20.25 โ ๐๐๐๐ = 3(20.25 โ 15) 4.5
Measures of Dispersion (Absolute and Relative)
Formula Calculation Answer
1 Range (R) ๐ฟ โ ๐ 48 โ 3 45
Co โ efficient of Range ๐ฟ โ ๐
๐ฟ + ๐ =
48โ3
48+3 0.8823
2 Quartile Deviation (๐ธ๐ซ) ๐3 โ ๐1
2
42 โ 3.75
2
19.125
Coefficient of Quartile Deviation ๐3 โ ๐1
๐3 + ๐1
42 โ 3.75
42 + 3.75
0.84
3 Mean Deviation (๐๐ทXฬ ) 1
๐โ|๐ โ Xฬ |
63
4
15.75
Co โ efficient of MD ๐๐ทXฬ
๐๐๐๐
15.75
20.25
0.778
4 Standard Deviation (๐) โ
โ(๐ โ Xฬ )2
๐ โ
1284.75
5
17.921
Or
โ
โ๐2
๐โ (
โ๐
๐)
2
โ2925
4โ (
81
4)
2
17.921
๐๐๐ (๐) ๐2 17.9212 321.16
๐ถ๐๐๐๐๐๐๐๐๐๐ก ๐๐ ๐ฃ๐๐๐๐๐ก๐๐๐, ๐ฃ๐๐(๐ฅ) ๐
Xฬ ร 100
17.921
20.25ร 100
88.49%
Working note
๐ฟ |๐ฟ โ ๏ฟฝฬ ๏ฟฝ| ๐ฟ โ ๐ (๐ฟ โ ๏ฟฝฬ ๏ฟฝ)๐ ๐ฟ๐
3 17.25 17.25 297.5625 9
6 14.25 14.25 203.0625 36
24 3.75 -3.75 14.0625 576
48 27.25 -27.75 770.0625 2304
Total 63 1284.75 2925
15.6
Question 2: Find Median, ๐ธ๐, ๐ธ๐,๐ถ๐, ๐ซ๐, ๐ท๐๐ for the observations: 1, 3, 6, 24, 48.
Answer:
Positional Average
Formula Calculations Answer
๐๐ ๐ ๐๐ง๐ ๐๐ (
๐ + 1
2)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 3๐๐ ๐๐๐ 6 + 0.5(24 โ 6) 6
๐1 ๐ ๐๐ง๐ ๐๐ (
๐ + 1
4)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 1.5๐กโ ๐๐๐ 1 + 0.5(3 โ 1) 2
1๐ ๐ก ๐๐๐ + 0.5 (2๐๐ ๐๐๐ โ 1๐ ๐ก ๐๐๐ )
๐3 ๐ ๐๐ง๐ ๐๐ (
3(๐ + 1)
4)
๐กโ
๐๐๐
๐๐๐ง๐ ๐๐ 4.5๐กโ ๐๐๐
4๐กโ ๐๐๐ + 0.5 (5๐กโ ๐๐๐ โ 4๐กโ ๐๐๐ )
24 + 0.5 (48 โ 24)
36 ๐75
๐ ๐๐ง๐ ๐๐ (75(๐ + 1)
100)
๐กโ
๐๐๐
๐6 ๐ ๐๐ง๐ ๐๐ (
6(๐ + 1)
8)
๐กโ
๐๐๐
๐ต๐๐๐: ๐3 = ๐6 = ๐75
๐ท7 ๐ ๐๐ง๐ ๐๐ (
7(๐ + 1)
10)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 4.2๐กโ ๐๐๐ 24 + 0.2(48 โ 24) ๐๐. ๐
4๐กโ ๐๐๐ + 0.2 (5๐กโ ๐๐๐ โ 4๐กโ ๐๐๐ )
Question 3: Discrete Frequency Distribution
x 10 11 12 13 14 15 16 17 18 19
f 8 15 20 100 98 95 90 75 50 30
Answer:
Measures of Averages
Formula Calculation Answer
1 Arithmetic Mean(xฬ ) โ๐๐
๐
8727
581 15.02
2 Geometric Mean(๐บ๐) Antilog (โ ๐ log ๐
๐) Antilog (
682.4203
581) 14.95
3 Harmonic Mean (๐ป๐) ๐
โ๐
๐
581
39.25 14.802
Working Note:
๐ฟ ๐ ๐๐ฟ ๐ฅ๐จ๐ ๐ฟ ๐ ๐ฅ๐จ๐ ๐ฟ ๐
๐ฟ
10 8 80 1.0000 8.0000 0.800
11 15 165 1.0414 15.6210 1.360
12 20 240 1.0792 21.5840 1.670
13 100 1300 1.1139 111.3900 7.690
15.7
14 98 1372 1.1461 112.3178 7.000
15 95 1425 1.1761 111.7295 6.330
16 90 1440 1.2041 108.3690 5.625
17 75 1275 1.2304 92.2800 4.411
18 50 900 1.2553 62.7650 2.780
19 30 570 1.2788 38.364 1.578
Total 581 8727 682.4203 39.25
Positional Average
Formula Calculations Answer Working Notes
๐๐ ๐ ๐๐ง๐ ๐๐ (๐ + 1
2)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 291๐ ๐ก ๐๐๐
(๐. ๐. ๐๐ > 291) 15
๐ฟ ๐ ๐๐
10 8 8
11 15 23
12 20 43
13 100 143
14 98 241
15 95 336
16 90 426
17 75 501
18 50 551
19 30 581
๐1 ๐ ๐๐ง๐ ๐๐ (1(๐ + 1)
4)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 145.5๐กโ ๐๐๐
(๐. ๐. ๐๐ > 145.5) 14
๐3 ๐ ๐๐ง๐ ๐๐ (3(๐ + 1)
4)
๐กโ
๐๐๐
๐๐๐ง๐ ๐๐ 436.5๐กโ ๐๐๐
(๐. ๐. ๐๐ > 436.5)
17
๐75 ๐ ๐๐ง๐ ๐๐ (75(๐ + 1)
100)
๐กโ
๐๐๐
๐6 ๐ ๐๐ง๐ ๐๐ (6(๐ + 1)
8)
๐กโ
๐๐๐
๐ต๐๐๐: ๐3 = ๐6 = ๐75
๐ท7 ๐ ๐๐ง๐ ๐๐ (7(๐ + 1)
10)
๐กโ
๐๐๐ ๐๐๐ง๐ ๐๐ 407.4๐กโ ๐๐๐
๐๐ (๐. ๐. ๐๐ > 407.4)
Mode: Since there is a sudden increase in frequency from 20 to 100, we obtain mode by Grouping
Table
Grouping Table The highest frequency total in each of the six
columns of the grouping table is identified and
analyzed (Tally marks)
Total
Tally
Mark
(1) (2) (3) (4) (5) (6)
๐ฟ ๐ (1) (2) (3) (4) (5) (6)
10 8 23
43
0
11 15 35
135
0
12 20 120
218
0
13 100 198
293
| | | 3
14 98 193
283
| | | | 4
15 95 185
260
| | | | 4
16 90 165
215
| | 2
17 75 125
155
| 1
18 50 80
0
19 30 0
15.8
Explanation to column
(๐) Original Frequency
(๐) grouping in โtwoโs
(๐) Leaving the first and grouping the
rest in โtwoโsโ
(๐) grouping in โthreeโsโ
(๐) Leaving the first and grouping in
โthreeโsโ
(๐) Leaving the first & second and
grouping in โthreeโsโ
Mode
Mode is ill-defined or bi-modal
(Since โ14โ and โ15โ occur equal number of times)
Hence, the empirical relation is used to arrive ๐๐
๐๐ ๐๐๐๐ โ ๐๐๐๐ = 3(๐๐๐๐ โ ๐๐๐๐๐๐)
15.02 โ ๐๐๐๐ = 3(15.02 โ 15)
14.96
Points to Ponder:
Under Location Method, Mode = 13 (as the highest frequency is 100)
Under Grouping Method, Mode is ill- defined.
But, Under Empirical Relationship, Mode = 14.96, which brings the issues an accuracy
Measures of Dispersion
Formula Calculation Answer
1 Range (R) ๐ฟ โ ๐ 19 โ 10 10
Co โ efficient of Range ๐ฟ โ ๐
๐ฟ + ๐
19 โ 10
19 + 10 0.31
2 Quartile Deviation (๐ธ๐ซ) ๐3 โ ๐1
2
17 โ 14
2 1.5
Coefficient of Quartile Deviation ๐3 โ ๐1
๐3 + ๐1
17 โ 14
17 + 14 0.0967
3 Mean Deviation (๐๐ทXฬ ) 1
๐โ|๐ โ Xฬ |
969.82
58.1 1.669
Co โ efficient of MD ๐๐ทXฬ
๐๐๐๐
1.669
15.02 0.111133
4 Standard Deviation (๐) โโ ๐(๐ โ Xฬ )2
๐ โ
2204.7628
581 3.80
๐๐๐ (๐) ๐ 2 3.802 14.44
๐ถ๐๐๐๐๐๐๐๐๐๐ก ๐๐ ๐ฃ๐๐๐๐๐ก๐๐๐, ๐ฃ๐๐(๐) ๐
xฬ ร 100
3.80
15.02ร 100 25.29%
15.9
Working Note:
for MD For SD
๐ฟ ๐ |๐ฟ โ ๏ฟฝฬ ๏ฟฝ| ๐|๐ฟ โ ๐| (๐ฟ โ ๏ฟฝฬ ๏ฟฝ) ๐(๐ฟ โ ๏ฟฝฬ ๏ฟฝ)๐
10 8 5 .02 40.16 -5 .02 201.6032
11 15 4.02 60.30 -4.02 242.4060
12 20 3.02 68.40 -3.02 182.4080
13 100 2.02 202.00 -2.02 81.6080
14 98 1.02 99.96 -1.02 101.9592
15 95 0.02 1.90 -0.02 0.0380
16 90 0.98 88.20 0.98 86.4360
17 75 1.98 143.50 1.98 294.0300
18 50 2.98 1.49 2.98 444.0200
19 36 3.98 119.40 3.98 570.2544
โ 581 969.82 2204.7628
Question 4: Continuous Frequency Distribution:
Marks 01-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100
Number of Students 3 7 13 17 12 10 8 8 6 6
Also verify the empirical relation
Answer:
Measures of Averages
Formula Calculation Answer
A.M. (Direct Method) โ ๐๐
๐
4375
90
48.61
A.M. (Short cut -Method) ๐ด + โ ๐๐
๐ร ๐ (A=45.5) 45.5 +
28
90ร 10
48.61
๐ด + โ ๐๐
๐ร ๐ (A = 55.5) 55.5 +
โ620
90ร 10
48.61
Geometric Mean, GM ๐ด๐๐ก๐๐๐๐ (
โ ๐ log ๐
๐) ๐ด๐๐ก๐๐๐๐
150.5439
90
47.07
Harmonic Mean, HM ๐
โ๐
๐
90
2.7905
32.25
15.10
Working Note:
Marks
(Class boundaries)
๐ ๐ ๐๐ ๐ =
๐ โ ๐๐. ๐
๐๐
๐๐ ๐ฅ๐จ๐ ๐ ๐ ๐ฅ๐จ๐ ๐ ๐
๐
0.5 โ 10.5 5.5 3 16.5 -4 -12 0.7404 2.2212 0.5454
10.5 โ 20.5 15.5 7 108.5 -3 -21 1.903 13.3210 0.4516
20.5 โ 30.5 25.5 13 331.5 -2 -26 1.4065 18.2845 0.5098
30.5 โ 40. 5 35.5 17 603.5 -1 -17 1.5502 26.3534 0.4789
40.5 โ 50.5 45.5 12 546.0 0 0 1.6580 19.8960 0.2637
50.5 โ 60.5 55.5 10 555.0 1 10 1.7443 17.4430 0.1801
60.5 โ 70.5 65.5 8 524.0 2 16 1.8162 14.5296 0.1221
70.5 โ 80.5 75.5 8 604.0 3 24 1.8779 15.0232 0.1060
80.5 โ 90.5 85.5 6 513.0 4 24 1.9320 11.5920 0.0701
90.5 โ 100.5 95.5 6 573.0 5 30 1.9800 11.8800 0.0628
Total 90 4375.0 28 150.5439 2.7905
Positional Average and Mode
Formula Calculation Answer
Working Note
๐๐ ๐ +
๐
2โ ๐
๐ร ๐ 40.5 +
45 โ 40
12ร 10 44.67
๐ฟ ๐ ๐๐
0.5โ10.5 3 3
10.5โ20.5 7 10
20.5โ30.5 13 23
30.5โ40. 5 17 40
40.5โ50.5 12 52
50.5โ60.5 10 62
60.5โ70.5 8 70
70.5โ80.5 8 78
80.5โ90.5 6 84
90.5โ100.5 6 90
๐1 ๐ +
1๐
4โ ๐
๐ร ๐ 20.5 +
22.5 โ 10
13ร 10 30.12
๐3 ๐ +
3๐
4โ ๐
๐ร ๐
60.5 +67.5 โ 62
8ร 10 67.38 ๐6 ๐ +
6๐
8โ ๐
๐ร ๐
๐75 ๐ +
75๐
100โ ๐
๐ร ๐
๐3 = ๐6 = ๐75 = 67.38
๐ท7 ๐ +
7๐
10โ ๐
๐ร ๐ 60.5 +
63 โ 62
8 ร 10 61.75
๐๐ ๐1 + (๐0 โ ๐โ1
2๐0 โ ๐โ1 โ ๐1
) ร ๐ถ 30.5 + (17 โ 13
2 ร 17 โ 13 โ 12) ร 10 34.94
๐ด๐ ๐๐๐๐๐ ๐๐ (๐๐. ๐ โ ๐๐. ๐), since 17 is the highest frequency
Graphical Method: Ogive Curves for Positional Average:
Marks Number
of Students
Less than ogive curve More than ogive curve
UCL < cf LCL >cf
15.11
0.5 โ 10.5 3 10.5 3 0.5 90 (= โ๐)
10.5 โ 20.5 7 20.5 10 10.5 87
20.5 โ 30.5 13 30.5 23 20.5 80
30.5 โ 40. 5 17 40. 5 40 30.5 67
40.5 โ 50.5 12 50.5 52 40.5 50
50.5 โ 60.5 10 60.5 62 50.5 38
60.5 โ 70.5 8 70.5 70 60.5 28
70.5 โ 80.5 8 80.5 78 70.5 20
80.5 โ 90.5 6 90.5 84 80.5 12
90.5 โ 100.5 6 100.5 90 (= โ๐) 90.5 6
Verification of Empirical relation:
Mean โ Mode = 3 (Mean - Median)
(i.e.,) 48.61 โ 34.94 = 3 (48.61 โ 44.67)
13.67 = 3 ( 4.006)
13.67 = 12.18, which is not true
Graphical Method
๐๐ = 35 (๐บ๐๐๐โ๐๐๐๐ ๐๐๐กโ๐๐)
๐1 = 30, ๐3 = 45 & ๐3 = 67
15.12
Measures of Dispersion
Formula Calculation Answer
1 Range (R) ๐ฟ โ ๐ 100 โ 1 99
Other-way 100.5 โ 0.5 100
Co โ efficient of Range ๐ฟ โ ๐
๐ฟ + ๐
100 โ 1
100 + 1 0.98
2 Quartile Deviation (๐ธ๐ซ) ๐3 โ ๐1
2
67.38 โ 30.12
2 18.63
Coefficient of Quartile Deviation ๐3 โ ๐1
๐3 + ๐1
67.38 โ 30.12
67.38 + 30.12 0.38
3 Mean Deviation (๐๐ทXฬ ) 1
๐โ|๐ โ Xฬ |
1843.54
90 20.48
Co โ efficient of MD ๐๐ทXฬ
๐๐๐๐
20.48
48.61 0.42
4 Standard Deviation (๐) โโ ๐(๐ โ Xฬ )2
๐ โ
53128.89
90 24.29
๐๐๐ (๐) ๐2 (24.29)2 590.49
๐ถ๐๐๐๐๐๐๐๐๐๐ก ๐๐ ๐ฃ๐๐๐๐๐ก๐๐๐, ๐ฃ๐๐(๐) ๐
Xฬ ร 100
24.29
48.61ร 100 25.29%
Working Notes
Marks
(Class boundaries) ๐ ๐ |๐ โ ๐| f|๐ โ ๏ฟฝฬ ๏ฟฝ| (๐ฟ โ ๐)๐ f (๐ฟ โ ๐)๐
0.5 โ 10.5 5.5 3 43.11 129.33 1858.4701 5575.4163
10.5 โ 20.5 15.5 7 33.11 231.77 1096.2721 7673.9047
20.5 โ 30.5 25.5 13 23.11 300.43 534.0721 6942.9373
30.5 โ 40. 5 35.5 17 13.11 222.87 171.8721 2921.8252
40.5 โ 50.5 45.5 12 3.11 37.32 9.6721 116.0652
50.5 โ 60.5 55.5 10 6.89 68.90 47.4733 474.7210
60.5 โ 70.5 65.5 8 16.89 135.12 285.2721 2282.1768
70.5 โ 80.5 75.5 8 26.89 215.12 723.0721 5784.5768
80.5 โ 90.5 85.5 6 36.89 221.34 1360.8721 8165.2326
90.5 โ 100.5 95.5 6 46.89 281.34 2198.6721 13192.0326
Total 90 1843.54 53128.889
15.13
PROPERTIES: (A) MEASURES OF AVERAGES / CENTRAL TENDENCY
Arithmetic Mean
Property 1: If all the observations assumed by a variable are constants, say k, then the AM is also k.
Illustration: Consider 2, 2, 2
Property Calculation Answer
Xฬ =๐ + ๐ + โฏ + ๐
๐= ๐
2 + 2 + 2
3 2
Property 2: (a) The algebraic sum of deviations of a set of observations from their AM is zero. And
(b) the sum of the square of the deviation taken from the Mean (Xฬ ) is always minimum compared to
the deviations taken from any other Assumed Mean (๐ด)
Illustration: Consider (X): 2, 3, 4
Property Formula Calculation Answer
Xฬ = โ ๐ 2 + 3 + 4
3 3
(a) โ(๐ โ Xฬ ) = 0
โ ๐(๐ โ Xฬ ) = 0 โ(๐ โ Xฬ ) (2 โ 3) + (3 โ 3) + (4 โ 3) 0
(b) โ(๐ โ Xฬ )2 โค โ(๐ โ ๐ด)2
โ(๐ โ Xฬ )2 (2 โ 3)2 + (3 โ 3)2 + (4 โ 3)2 2
โ(๐ โ ๐ด)2
๐โ๐๐๐ ๐ด = 4
(2 โ 4)2 + (3 โ 4)2 + (4 โ 4)2 5
Property 3: AM is affected due to a change of origin (+/โ) and / or scale (ร/รท)
i.e., If ๐ฆ = ๐ + ๐๐ฅ, then the AM of y is given by yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ (where a is change of origin and b is change
of scale)
Illustration: Consider (๐) = 2, 3, 4,
Formula Calculation Answer ๏ฟฝฬ ๏ฟฝ =โ ๐
๐= ๐ + ๐๐
1 ๐ฟ = ๐, ๐, ๐, ๏ฟฝฬ ๏ฟฝ =โ ๐ฟ
๐
๐ + ๐ + ๐
๐ 3
2 ๐ = 4, 5, 6, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
4 + 5 + 6
3 5
Change of Origin (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ + 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 2 + 1 ร3 5
3 ๐ = 0, 1, 2, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
0 + 1 + 2
3 1
Change of Origin (๐ = โ2)
๐ต๐๐๐๐ ๐ = ๐ โ 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ โ2 + 1 ร3 1
4 ๐ = 4, 6, 8, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
4 + 6 + 8
3 6
Change of Scale (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ ร 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 0 + 2 ร3 6
15.14
5 ๐ = 1, 1.5, 2, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
1 + 1.5 + 2
3 1.5
Change of Scale (๐ =1
2)
๐ต๐๐๐๐ ๐ = ๐ ร1
2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 0 +
1
2ร3 1.5
6 ๐ = 7, 9, 11, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
7 + 9 + 11
3 9 Change of Origin and
change of scale
(๐ = 3)&(๐ = 2) ๐ต๐๐๐๐ ๐ = 3 + 2 ร ๐ yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 3 + 2 ร3 9
Property 4: If there are two groups containing ๐1 and ๐2 observations and ๏ฟฝฬ ๏ฟฝ1 and ๏ฟฝฬ ๏ฟฝ2 as the respective
arithmetic means, then the combined AM is given by (๏ฟฝฬ ๏ฟฝ12) =๐1๏ฟฝฬ ๏ฟฝฬ 1+๐2๏ฟฝฬ ๏ฟฝฬ 2
๐1+๐2
Illustration Combined mean Calculation Answer
Group 1 Group II
๐1 = 5 ๐2 = 15
๏ฟฝฬ ๏ฟฝ1 = 9 ๏ฟฝฬ ๏ฟฝ2 = 5
๏ฟฝฬ ๏ฟฝ12 =๐1๏ฟฝฬ ๏ฟฝ1 + ๐2๏ฟฝฬ ๏ฟฝ2
๐1 + ๐2
(5 ร 9) + (15 ร 5)
5 + 15 6
Points to Ponder:
1 In the case of โnโ number of groups, Combined mean (๏ฟฝฬ ๏ฟฝ1โฆ๐) =โ๐๐๏ฟฝฬ ๏ฟฝฬ ๐
โ๐๐
2 If sizes of the group are same, then the combined Mean is the average of the group means
Explanation: If ๐1= ๐2 =n, then ๏ฟฝฬ ๏ฟฝ1+2 =๐๏ฟฝฬ ๏ฟฝฬ 1+๐๏ฟฝฬ ๏ฟฝฬ 2
๐+๐=
๐(๏ฟฝฬ ๏ฟฝฬ 1+๏ฟฝฬ ๏ฟฝฬ 2)
2๐=
๏ฟฝฬ ๏ฟฝฬ 1+ ๏ฟฝฬ ๏ฟฝฬ 2
2
Illustration
Formula Calculation Answer
1 ๐1 = 2, 3, 4, ๏ฟฝฬ ๏ฟฝ1 =โ ๐1
๐
2 + 3 + 4
3 3
2 ๐2 = 4, 5, 6, ๏ฟฝฬ ๏ฟฝ2 =โ ๐2
๐
4 + 5 + 6
3 5
3 ๏ฟฝฬ ๏ฟฝ1+2 =๐๏ฟฝฬ ๏ฟฝ1 + ๐๏ฟฝฬ ๏ฟฝ2
๐ + ๐
3 ร 3 + 3 ร 5
3 + 3 4
๏ฟฝฬ ๏ฟฝ1+2 =๐(๏ฟฝฬ ๏ฟฝ1 + ๏ฟฝฬ ๏ฟฝ2)
2๐
3(3 + 5)
2 ร 3 4
๏ฟฝฬ ๏ฟฝ1+2 =๏ฟฝฬ ๏ฟฝ1 + ๏ฟฝฬ ๏ฟฝ2
2
3 + 5
2 4
3 If the averages are same, then the combined mean is the average itself
Explanation: If ๏ฟฝฬ ๏ฟฝ1 = ๏ฟฝฬ ๏ฟฝ2 = ๏ฟฝฬ ๏ฟฝ12
๏ฟฝฬ ๏ฟฝ12 =๐1๏ฟฝฬ ๏ฟฝ + ๐2๏ฟฝฬ ๏ฟฝ
๐1 + ๐2
=๏ฟฝฬ ๏ฟฝ(๐1 + ๐2)
๐1 + ๐2
Illustration
Formula Calculation Answer
1 ๐1 = 2, 3, 4, ๏ฟฝฬ ๏ฟฝ1 =โ ๐1
๐
2 + 3 + 4
3 3
2 ๐2 = 4, 2, ๏ฟฝฬ ๏ฟฝ2 =โ ๐2
๐
4 + 2
2 3
15.15
3 ๏ฟฝฬ ๏ฟฝ1+2 =๐๏ฟฝฬ ๏ฟฝ1 + ๐๏ฟฝฬ ๏ฟฝ2
๐ + ๐
3 ร 3 + 2 ร 3
3 + 2 3
๏ฟฝฬ ๏ฟฝ(๐1 + ๐2)
๐1 + ๐2
3(3 + 2)
2 + 3 3
๏ฟฝฬ ๏ฟฝ1+2 = ๏ฟฝฬ ๏ฟฝ1 = ๏ฟฝฬ ๏ฟฝ2 3
Geometric Mean
Property 1: Transformation in terms of log function
๐บ๐ = ๐ด๐๐ก๐๐๐๐ (1
๐โ ๐๐๐ ๐ฅ) ๐๐ ๐๐๐ ๐บ๐ =
1
๐โ ๐๐๐ ๐ฅ
Property 2: If all the observations assumed by a variable are constants, say ๐ > 0, then the GM of the
observations is also K.
Property Illustration Calculation Answer
(๐ ร ๐ ร โฆ .ร ๐)1 ๐โ = ๐ Consider: 2, 2, 2 = (2 ร 2 ร 2)1 3โ 2
Property 3: GM of the product of two variables is the product of their GMโs.
Property 4: GM of the ratio of two variables is the ratio of the GMโs of the two variables.
Illustration Formula Calculation Answer
๐ = 3, 6, 12 GM = (๐1 ร ๐2 ร โฆ ร ๐)1
๐ (3 ร 6 ร 12)1 3โ 6
๐ = 1, 2, 4 (1 ร 2 ร 4)1 3โ 2
๐ = 3, 12, 48 (3 ร 12 ร 48)1 3โ 12
Property 3 Being ๐ = ๐ ร ๐ GM๐ = GM๐ ร GM๐ 6 ร 2 12
๐ = 3
1,6
2,12
4 (3 ร 3 ร 3)1 3โ 3
Property 4 Being ๐ =๐
๐ ๐บ๐๐ง =
GM๐
GM๐
6
2 3
Harmonic Mean:
Property 1: If all the observations taken by a variable are constants, say k, then the HM of the
observations is also k.
Property Illustration Calculation Answer
๐1
๐+
1
๐+ โฆ . +
1
๐
= ๐ ๐ = 2, 2, 2 31
2+
1
2+
1
2
2
Property 2: If there are two groups containing ๐๐ and ๐๐ observations and ๐ฟ๐ and ๐ฟ๐ as the
respective Harmonic Means, then the combined HM is given by (๏ฟฝฬ ๏ฟฝ๐๐) = ๐๐+๐๐๐๐๏ฟฝฬ ๏ฟฝ๐
+ ๐๐๏ฟฝฬ ๏ฟฝ๐
Illustration Combined H.M. Calculation Answer
Group 1 Group II ๏ฟฝฬ ๏ฟฝ๐๐ =๐1 + ๐2๐1
๏ฟฝฬ ๏ฟฝฬ 1+
๐2
๏ฟฝฬ ๏ฟฝฬ 2
15 + 1015
3+
10
2
3.125
15.16
๐1 = 15 ๐2 = 10
๏ฟฝฬ ๏ฟฝ1 = 3 ๏ฟฝฬ ๏ฟฝ2 = 2
Median:
Property 1: If x and y are two variables, to be related by ๐ = ๐ + ๐๐ for any two constants a and b,
then the median of y is given by ๐๐๐= ๐ + ๐๐๐๐
(i.e., Median is affected due to a change of origin (+/โ) and / or scale (ร/รท))
Illustration: Consider (๐) = 2, 3, 4,
Formula Calculation Answer ๐๐ด๐= ๐ + ๐๐ฟ๐ด๐
1 ๐ = 2, 3, 4, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 3
2 ๐ = 4, 5, 6, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 5 Change of Origin (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ + 2 ๐๐๐= ๐ + ๐๐๐๐
2 + 1 ร3 5
3 ๐ = 0, 1, 2, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 1 Change of Origin (๐ = โ2)
๐ต๐๐๐๐ ๐ = ๐ โ 2 ๐๐๐= ๐ + ๐๐๐๐
โ2 + 1 ร3 1
4 ๐ = 4, 6, 8, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 6 Change of Scale (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ ร 2 ๐๐๐= ๐ + ๐๐๐๐
0 + 2 ร3 6
5 ๐ = 1, 1.5, 2, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 1.5
Change of Scale (๐ =1
2)
๐ต๐๐๐๐ ๐ = ๐ ร1
2 ๐๐๐
= ๐ + ๐๐๐๐ 0 +
1
2ร3 1.5
6 ๐ = 7, 9, 11, ๏ฟฝฬ ๏ฟฝ๐๐= (
๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 9 Change of Origin and
change of scale
(๐ = 3)&(๐ = 2) ๐ต๐๐๐๐ ๐ = 3 + 2 ร ๐ ๐๐๐= ๐ + ๐๐๐๐
3 + 2 ร3 9
Property 2: For a set of observations, the sum of absolute deviations is minimum when the deviations
are taken from the median.
Illustration: Consider (X): 0.5, 3, 4
Calculation Answer Property
๐๐ = (๐ + 1
2)
๐กโ
๐๐๐ (3 + 1
2)
๐กโ
๐๐๐ 3
๏ฟฝฬ ๏ฟฝ =โ ๐
๐
0.5 + 3 + 4
3 2.5
(a) โ |๐ โ Xฬ | |0.5 โ 2.5| + |3 โ 2.5| + |4 โ 2.5| 4
(๐) < (๐)
(b) โ |๐ โ ๐๐| |0.5 โ 3| + |3 โ 3| + |4 โ 3| 3.5
15.17
Mode:
Property 1: If ๐ = ๐ + ๐๐, then ๐๐๐= ๐ + ๐๐๐๐
(i.e., Mode is affected due to a change of origin (+/โ) and / or scale (ร/รท))
Illustration: Consider (๐) = 2, 3, 3, 4
Formula Calculation Answer ๐๐ด๐= ๐ + ๐๐ฟ๐ด๐
1 ๐ = 2, 3, 3, 4, Most usual 3
2 ๐ = 4, 5, 5, 6, 5 Change of Origin (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ + 2 ๐๐๐= ๐ + ๐๐๐๐
2 + 1 ร3 5
3 ๐ = 0, 1, 1, 2, Most usual 1 Change of Origin (๐ = โ2)
๐ต๐๐๐๐ ๐ = ๐ โ 2 ๐๐๐= ๐ + ๐๐๐๐
โ2 + 1 ร3 1
4 ๐ = 4, 6, 6, 8, Most usual 6 Change of Scale (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ ร 2 ๐๐๐= ๐ + ๐๐๐๐
0 + 2 ร3 6
5 ๐ = 1, 1.5, 1.5, 2, Most usual 1.5
Change of Scale (๐ =1
2)
๐ต๐๐๐๐ ๐ = ๐ ร1
2 ๐๐๐
= ๐ + ๐๐๐๐ 0 +
1
2ร3 1.5
6 ๐ = 7, 9, 9, 11, Most usual 9 Change of Origin and
change of scale
(๐ = 3)&(๐ = 2) ๐ต๐๐๐๐ ๐ = 3 + 2 ร ๐ ๐๐๐
= ๐ + ๐๐๐๐ 3 + 2 ร3 9
(B) MEASURES OF DISPERSION: PROPERTY
Property Measure / Explanation
1 All the observations assumed by a variable are constant,
then measure of dispersion = 0
Range (R) = 0
Mean Deviation (MD) = 0
Standard Deviation (s) = 0
Illustration: Consider (๐ฟ): 2, 2, 2
Formula Calculation Answer
๏ฟฝฬ ๏ฟฝ =โ ๐ฟ
๐
2 + 2 + 2
3 2
Range = L โ S 2 โ 2
0 ๐๐ทXฬ =
1
๐โ|๐ โ Xฬ |
|2 โ 2| + |2 โ 2| + |2 โ 2|
3
๐๐ท = โโ(๐ โ Xฬ )2
๐ โโ
(๐ โ 2)2
3
2 Affected due to change of Scale, but not of origin ๐ ๐ฆ = 0 + |๐| ร ๐ ๐ฅ
๐๐ทyฬ = 0 + |๐| ร MDxฬ
๐ ๐ฆ = 0 + |๐| ร ๐ ๐ฅฬ
3 Mean deviation takes its minimum value ๐๐ท๐๐=
1
๐โ|๐ โ ๐๐| is minimum
15.18
when A = Median
4 Combined SD ๐ 12 = โ
๐1๐12 + ๐2๐2
2 + ๐1๐12 + ๐2๐2
2
๐1 + ๐2
where ๐1 = ๏ฟฝฬ ๏ฟฝ1 โ ๏ฟฝฬ ๏ฟฝ12 and ๐2 = ๏ฟฝฬ ๏ฟฝ2 โ ๏ฟฝฬ ๏ฟฝ12
Note: If ๏ฟฝฬ ๏ฟฝ1 = ๏ฟฝฬ ๏ฟฝ2 , then ๏ฟฝฬ ๏ฟฝ1 = ๏ฟฝฬ ๏ฟฝ2 = ๏ฟฝฬ ๏ฟฝ12
Then ๐1 = 0 & ๐2 = 0
โด ๐ 12 = โ๐1๐1
2 + ๐2๐22
๐1 + ๐2
Illustration Calculation Answer
Group I Group II
๐1 = 5 ๐2 = 15
๏ฟฝฬ ๏ฟฝ1 = 9 ๏ฟฝฬ ๏ฟฝ2 = 5
๐ 1 = 0.8 ๐ 2 = 0.5
๏ฟฝฬ ๏ฟฝ12 = 6
๐ 12 = โ5 ร (0.8)2 + (15 ร (0.5)2) + (5 ร 32) + (15 ร (โ1)2)
5 + 15
๐1 = ๏ฟฝฬ ๏ฟฝ1 - ๏ฟฝฬ ๏ฟฝ12 = 9 โ 6 = 3
๐2 = ๏ฟฝฬ ๏ฟฝ2 โ ๏ฟฝฬ ๏ฟฝ12 = 5 โ 6 = โ1
1.83
Problem for SD under Change of scale and origin
Formula Calculation Answer ๏ฟฝฬ ๏ฟฝ =โ ๐
๐= ๐ + ๐๐
1 ๐ฟ = ๐, ๐, ๐, ๏ฟฝฬ ๏ฟฝ =โ ๐ฟ
๐
๐ + ๐ + ๐
๐ 3
๐ ๐ = ๐ฟ โ ๐ 4 โ 2 2
๐๐ทxฬ =โ|๐ โ Xฬ |
๐
โ|๐ โ 3|
3
2
3
๐ ๐ = โโ(๐ โ Xฬ )2
๐ โ
โ(๐ โ 3)2
3 0.82
2 ๐ = 4, 5, 6, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
4 + 5 + 6
3 5
Change of Origin (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ + 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 2 + 1 ร3 5
๐ ๐ = ๐ฟ โ ๐ 6โ4 2
๐๐ทYฬ =โ|๐ โ Yฬ |
๐
โ|๐ โ 5|
3
2
3
๐ ๐ = โโ(๐ โ Yฬ )2
๐ โ
โ(๐ โ 3)2
3 0.82
3 ๐ = 0, 1, 2, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
0 + 1 + 2
3 1
Change of Origin (๐ = โ2)
๐ต๐๐๐๐ ๐ = ๐ โ 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ โ2 + 1 ร3 1
๐ ๐ = ๐ฟ โ ๐ 2 โ 0 2
๐๐ทYฬ =โ|๐ โ Yฬ |
๐
โ|๐ โ 1|
3
2
3
15.19
๐ ๐ = โโ(๐ โ Yฬ )2
๐ โ
โ(๐ โ 1)2
3 0.82
4 ๐ = 4, 6, 8, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
4 + 6 + 8
3 6
Change of Scale (๐ = 2)
๐ต๐๐๐๐ ๐ = ๐ ร 2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 0 + 2 ร3 6
๐ ๐ = ๐ฟ โ ๐ 8 โ 4 4
๐๐ทYฬ =โ|๐ โ Yฬ |
๐
โ|๐ โ 6|
3
4
3
๐ ๐ = โโ(๐ โ Yฬ )2
๐ โ
โ(๐ โ 6)2
3 1.64
5 ๐ = 1, 1.5, 2, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
1 + 1.5 + 2
3 1.5
Change of Scale (๐ =1
2)
๐ต๐๐๐๐ ๐ = ๐ ร1
2 yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 0 +
1
2ร3 1.5
๐ ๐ = ๐ฟ โ ๐ 2 โ 1 1
๐๐ทYฬ =โ|๐ โ Yฬ |
๐
โ|๐ โ 1.5|
3
1
3
๐ ๐ = โโ(๐ โ Yฬ )2
๐ โ
โ(๐ โ 1.5)2
3 0.41
6 ๐ = 7, 9, 11, ๏ฟฝฬ ๏ฟฝ =โ ๐
๐
7 + 9 + 11
3 9 Change of Origin and
change of scale
(๐ = 3)&(๐ = 2) ๐ต๐๐๐๐ ๐ = 3 + 2 ร ๐ yฬ = ๐ + ๐๏ฟฝฬ ๏ฟฝ 3 + 2 ร3 9
๐ ๐ = ๐ฟ โ ๐ 11 โ 7 4
๐๐ทxฬ =โ|๐ โ Xฬ |
๐
โ|๐ โ 9|
3
4
3
๐ ๐ = โโ(๐ โ Xฬ )2
๐ โ
โ(๐ โ 9)2
3 0.41
Coefficient of Variation (CV): ๐ถ๐ =๐
๏ฟฝฬ ๏ฟฝฬ ร 100
Illustration Calculation Comparison
Group 1 Group II
๏ฟฝฬ ๏ฟฝ1 = 9 ๏ฟฝฬ ๏ฟฝ2 = 5
๐ 1 = 0.8 ๐ 2 = 0.5
๏ฟฝฬ ๏ฟฝ12 = 6
๐ถ๐(๐ผ) =0.8
9 ร 100 = 8.88%
๐ถ๐(๐ผ๐ผ) =0.5
5ร 100 = 10%
๐ถ๐(๐ผ) = 8.88% < ๐ถ๐(๐ผ๐ผ) = 10%
More Stable
More Consistent
Less Variable
Less Dispersed
Less Stable
Less Consistent
More Variable
More Dispersed
EXTRA PROBLEMS
Comparison between Arithmetic Mean and Geometric Mean
Question 1: Find the average rate of return.
15.20
Year 1 2 3
Rate of Return (r %) 10% 60% 20%
Answer: The average rate of return
Formula Calculation Answer
GM G = (๐1 ร ๐2 ร โฆ ร ๐๐)1
๐ (1.10 ร 1.60 ร 1.20)1 3โ 1.283 ๐๐ 128.3% ๐๐ 28.3%
AM Xฬ =
โ ๐
๐
1.10 + 1.60 + 1.20
3
1.3 ๐๐ 130% ๐๐ 30%
which is not possible
Comparison between Arithmetic Mean and Harmonic Mean
Question 2: An aeroplane covered a distance of 800 miles with four different speeds of 100, 200, 300
and 400 m/p.h for the first, second, third and fourth quarter of the distance. Find the average speed in
miles per hour.
Answer: The average speed is given by the H.M. of the given set of data.
Formula Calculation Answer
H M ๐ป๐ =๐
โ1
๐
41
100+
1
200+
1
300+
1
400
192 m/p.h
AM Xฬ =
โ ๐
๐
100 + 200 + 300 + 400
4
250 m/p.h,
which is not true
Combined Mean
Question 3: Two groups of students reported mean weights of 160 kg and 150 kg respectively. Find
out, when the weight of both the groups together be 155 kg?
Answer:
Given Data Formula Calculation Answer
Group I Group II
Number ๐1 ๐2
Mean (kg.) Xฬ 1 = 160 Xฬ 2 = 150
Combined Mean: Xฬ 12 = 155kg
Xฬ 12 =๐1Xฬ 1 + ๐2๏ฟฝฬ ๏ฟฝ2
๐1 + ๐2
155 =160๐1 + 150๐2
๐1 + ๐2
155๐1 + 155๐2 = 160๐1 + 150๐2
๐1 = ๐2
Question 4: Show that for any two numbers a and b, standard deviation is given by |๐โ๐|
2
Answer: For two numbers a and b, AM is given by Xฬ =๐+๐
2
The variance is =โ(๐๐ โ Xฬ )2
2
=(๐ โ
๐+๐
2)
2
+ (๐ โ ๐+๐
2)
2
2=
(๐โ๐)
4
2
+ (๐โ๐)2
4
2=
(๐ โ ๐)2
4 โน ๐ =
|๐ โ ๐|
4
(The absolute sign is taken, as SD cannot be negative).
Question 5: Prove that for the first n natural numbers, ๐๐ โ๐2โ 1
12 .
Answer: for the first n natural numbers AM is given by
15.21
Xฬ =1 + 2 + โฆ โฆ โฆ + ๐
๐=
๐(๐ + 1)
2๐=
๐ + 1
2
โด ๐๐ท = โโ ๐๐
2
๐โ Xฬ 2 = โ
12 + 22 + 32 โฆ โฆ . . +๐2
๐โ (
๐ + 1
2)
2
โ๐(๐ + 1)(2๐ + 1)
6๐โ (
๐ + 1
2)
2
= โ(๐ + 1)(2๐ + 1)
6โ (
๐ + 1
2)
2
โ(๐ + 1)(2๐ + 1)
6โ
๐ + 1
2ร
๐ + 1
2= โ(๐ + 1) (
(2๐ + 1)
6โ
๐ + 1
4)
โ(๐ + 1)(4๐ + 2 โ 3๐ โ 3)
12= โ
๐2 โ 1
12
Thus, SD of first n natural numbers is SD = โ๐2 โ 1
12
15.22
COMPARISON BETWEEN MEASURES OF CENTRAL TENDENCY N
o
Mea
sure
s
Ari
thm
etic
Mea
n
Geo
met
ric
Mea
n
Har
mo
nic
Mea
n
Med
ian
Mo
de
Ran
ge
Qu
arti
le
Dev
iati
on
Mea
n
Dev
iati
on
Sta
nd
ard
Dev
iati
on
1 Well defined Yes Yes Yes Yes
No (when the
number of
observations is
small, then use
Empirical
Relationship)
Yes Yes A may be
Xฬ , ๐๐, ๐๐ Yes
2 Easy to calculate &
simple to understand Yes No No Yes
Location Method,
but not Grouping
method
Yes Yes Yes No
3 Based on all the
items Yes
Yes (but able
to find only
for Positive
Values)
Yes
(ONLY
positive
values
and no
โ0โ)
No No No No Yes Yes
4
capable of further
mathematical
treatment
Yes
Yes (Useful
for
calculation of
Index
Numbers)
Yes
Yes (but only in
Mean Deviation,
no combined
Median)
No
No (But in case
of Quality
control and stock
market
fluctuations)
No
No (Useful for
Economists and
Businessmen and
in public reports)
Yes
5 Good basis for
comparison Yes Not much Yes
6 Necessary for
arrange of data No No No Yes No ------Not on Discussion-----
7 Affected by extreme
values Yes
Yes (Not
much
compared to
Yes No No Yes No Less than SD Yes
15.23
AM)
8
Not Precise โ Mis-
leading impressions
(E.g. Average
number of persons is
1.5 which is not
possible)
No
No No
Yes (except
when Median
lies in between
two values)
Yes (except on
continuous series) ------Not on Discussion-----
9 Location (Inspection)
Method No No No
Yes (on
arrangement) Yes ------Not on Discussion-----
10 Graphical Method Yes (using Ogive
Curves) ------Not on Discussion-----
11
Calculated in the
case of open end
class intervals
No No No Yes Yes No Yes Based on โAโ No
12
Affected by
sampling
fluctuations
No
(least) No No Yes Yes Yes Yes Yes
Less
affected
13
Affected by Change
of origin Yes Yes Yes Yes Yes No No No No
Affected by Change
of Scale Yes Yes Yes Yes Yes Yes Yes Yes Yes
15.24
Explanations to Formulae:
1. Geometric Mean
Logarithmic formulae of Geometric Mean
Individual Observation Discrete Continuous
GM = โ๐ฅ1 ร ๐ฅ2 ร โฆ .ร ๐ฅ๐๐
log ๐บ. ๐ = log โ๐ฅ1 ร ๐ฅ2 ร โฆ .ร ๐ฅ๐๐
= 1
๐log(๐ฅ1 ร ๐ฅ2 ร. . .ร ๐ฅ๐)
= 1
๐(log ๐ฅ1 + log ๐ฅ2 + โฆ . log ๐ฅ๐)
= 1
๐โ log ๐ฅ
GM = Anti log (1
๐โ log ๐ฅ)
GM = โ๐ฅ1๐1 ร ๐ฅ2
๐2 ร โฆ . ๐ฅ๐๐๐
๐
log ๐บ. ๐ = log โ๐ฅ1๐1 ร ๐ฅ2
๐2 ร โฆ . ๐ฅ๐๐๐
๐
= 1
๐[(log ๐ฅ1
๐1 ร ๐ฅ2๐2 ร โฆ . ๐ฅ๐
๐๐)]
= 1
๐[log ๐ฅ1
๐1 + log ๐ฅ2๐2 + โฆ . log ๐ฅ๐
๐๐]
= 1
๐[๐1 log ๐ฅ1 + ๐2 log ๐ฅ2 + โฏ ๐๐ log ๐ฅ๐]
= 1
๐โ ๐ log ๐ฅ
GM = Antilog 1
๐โ ๐ log ๐ฅ
GM = โ
๐1๐1 ร ๐2
๐2 ร
โฆ .ร ๐๐๐๐
๐
log ๐บ. ๐ = log โ
๐1๐1 ร ๐2
๐2 ร
โฆ ร ๐๐๐๐
๐
= 1
๐[(log ๐1
๐1 ร ๐2๐2 ร โฆ ร ๐๐
๐๐)]
= 1
๐[log ๐1
๐1 + log ๐2๐2 + โฆ . log ๐๐
๐๐]
= 1
๐[๐1 log ๐1 + ๐2 log ๐2 + โฏ ๐๐ log ๐๐]
= 1
๐โ ๐ log ๐
GM = Antilog
1
๐โ ๐ log ๐
15.25
Standard Deviation:
๐ = โโ(๐ โ Xฬ )2
๐
โ(๐ โ Xฬ )2 = โ[๐2 โ 2๐Xฬ + Xฬ 2]
โ(๐ โ Xฬ )2 = โ ๐2 โ โ(2๐Xฬ ) + โ Xฬ 2
โ(๐ โ Xฬ )2 = โ ๐2 โ 2Xฬ โ ๐ + ๐Xฬ 2
โ(๐ โ Xฬ )2 = โ ๐2 โ 2โ ๐
๐โ ๐ + ๐.
โ ๐
๐.โ ๐
๐
โ(๐ โ Xฬ )2 = โ ๐2 โ 2(โ ๐)2
๐+
(โ ๐)2
๐
โ(๐ โ Xฬ )2 = โ ๐2 โ 2(โ ๐)2
๐+
(โ ๐)2
๐
โ(๐ โ Xฬ )2 = โ ๐2 โ(โ ๐)2
๐(2 โ 1)
โ(๐ โ Xฬ )2
๐=
โ ๐2 โ(โ ๐)2
๐
๐
โ(๐ โ Xฬ )2
๐=
๐ โ ๐2โ(โ ๐)2
๐
๐=
โ ๐2
๐โ (
โ ๐
๐)
2
=โ ๐2
๐โ Xฬ 2
15.26
Graphical Method
Weighted Average:
1. Calculate goodwill using weighted average method:
Profit 20,000 10,000 (7000)
Weight 3 2 1
Missing Frequency:
1. Given N = 581 and Mean = 15. Find the missing frequencies.
x 10 11 12 13 14 15 16 17 18 19
f 8 15 x 100 98 95 y 75 50 30
2. Given Mean = 47, Median = 45, Mode = 35 and N= 90. Find the missing frequencies.
Marks 01-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100
Number of Students 3 7 x 17 12 y 8 8 6 6