1.0 STATISTICS AND PROBABILITY 1.2 Measure of Central ...
Transcript of 1.0 STATISTICS AND PROBABILITY 1.2 Measure of Central ...
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 1
1.2 Measure of Central Tendency and Dispersion1.0 STATISTICS AND PROBABILITY
1.2.1 Calculate mean, median and mode for ungrouped data
1.2.2 Calculate mean, median and mode for grouped data by using formula
1.2.3 Calculate median and mode for grouped data by using graph
1.2.5 Calculate quartiles, deciles and percentiles for grouped data by
using formula and graph
1.2.4 Calculate mean deviation, variance, standard deviation for ungrouped
and grouped data
1
2
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 2
1.2.1 Calculate mean, median and mode for ungrouped data
MEAN, MEDIAN AND MODE FOR UNGROUPED DATA
Mean : The mean defined as the sum of all values in the set of data divided by the
number of values.
Mean, where: ฯ๐ฅ = sum of all the data
N = number of values in the set of data
If a set of data is presented in the form of a frequency distribution table
(either grouped or ungrouped data), then the mean of the set of data is
defined as below.
ฯ๐๐ฅ = sum of the value of frequency รmidpointwhere:
ฯ๐ = sum of frequency
3
4
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 3
Example 1:
1. Find the mean of the following data:
76 74 65 58 68 73
Example 2:
Solution:
x 1 2 3 4 5
f 20 43 20 15 12 โf = 110
fx 20 86 60 60 60 โfx = 286
5
6
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 4
Transfer the scores 1,2,2,3,3,3,3,4,4,5 into the table and the find the mean.
Example 3:
Marks
(x)
Frequency
(f)
fx
1 1 1
2 2 4
3 4 12
4 2 8
5 1 5
N = 10 30 =fx
๐๐๐๐, าง๐ฅ =ฯ ๐๐ฅ
๐
าง๐ฅ =30
10
าง๐ฅ = 3
Median: If the values of set of data are arranged in order of magnitude
(ascending or descending), the value of the middle is known as the
median of the set of data.
A set of data having n values that are arranged in order is given:
For data with odd number of values: For data with even number of values:
๐
2and
๐
2+ 1
๐ + 1
2
7
8
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 5
Example 4:
Determine the median of each of the following sets of data.
Solution:
Example 5:
Determine the median of each of the following sets of data.
Solution:
๐
2=
8
2= 4๐กโ and
8
2+ 1 = 5th
9
10
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 6
Example 6:
Determine the median of each of the following sets of data.
Solution:
Total number of values = 25
Mode: The mode of a set of data is,
the value that has the highest frequency
the value that appears the most number
of times in the set of data
the most commonly occurring value in a
set of data
11
12
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 7
Example 7:
Determine the mode of each of the following sets of data.
Tutorial 1.2.1 :
1. Find the mode, median and mean for the data given :
a) 5, 7, 4, 8, 2, 5, 9
b) 2, 3, 1, 2, 6, 8, 9, 3, 2, 3
c)
d)
13
14
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 8
1.2.2 Calculate mean, median and mode for grouped data by using formula
Mean of grouped data
To find the mean of the data, we should use the following formula:
ฯ๐๐ฅ = sum of the value of frequency รmidpointwhere:
ฯ๐ = sum of frequency
Example 8: Find the mean of mark of the student.
Mark Frequency (f)1 โ 20 2
21 โ 40 4
41 โ 60 12
61 โ 80 38
81 โ 100 25
101 โ 120 11
121 - 140 8
15
16
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 9
Solution :
Mark f x fx
1 โ 20 2 10.5 21
21 โ 40 4 30.5 122
41 โ 60 12 50.5 606
61 โ 80 38 70.5 2679
81 โ 100 25 90.5 2262.5
101 โ 120 11 110.5 1215.5
121 - 140 8 130.5 1044
โf =100 โfx= 7950
Using Formula,
๐๐๐๐๐๐,๐ = ๐ฟ๐ +
๐2โ ๐น
๐๐๐ถ
17
18
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 10
Example 9: Find the median mark of the student.
Mark Frequency (f)25 โ 35 5
36 โ 46 8
47 โ 57 14
58 โ 68 11
69 โ 79 7
80 โ 90 5
Solution :
Mark f F
25 โ 35 5 5
36 โ 46 8 13
47 โ 57 14 27
58 โ 68 11 38
69 โ 79 7 45
80 โ 90 5 50
Determine the class which median lies:
19
20
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 11
Therefore, median of the mark,
Mode of grouped data
where :
๐ด๐๐ ๐, ๐ฟ๐๐ +๐1
๐1 + ๐2๐ถ
๐ฟ๐๐ = Lower boundary of the class in which the mode lies๐1 = Difference between the frequency of the mode class and the class before it๐2 = Difference between the frequency of the mode class and the class after it๐ถ = ๐ถ๐๐๐ ๐ ๐ ๐๐ง๐
21
22
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 12
Mark Frequency (f)25 โ 35 536 โ 46 847 โ 57 1458 โ 68 1169 โ 79 780 โ 90 5
Example 10: Find the mode mark of the student.
Solution :
Determine the modal class: 47-57 โ modal class (has the highest frequency)
23
24
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 13
Find the mean, median, and mode of the following grouped data.
Example 11 :
Marks Frequency (f)
10 โ 14 2
15 โ 19 3
20 โ 24 8
25 โ 29 5
30 - 34 2
Solution :
Marks f F xClass
boundariesfx
10 โ 14 2 2 12 9.5 โ 14.5 24
15 โ 19 3 5 17 14.5 โ 19.5 51
20 โ 24 8 13 22 19.5 โ 24.5 176
25 โ 29 5 18 27 24.5 โ 29.5 135
30 - 34 2 20 32 29.5 โ 34.5 64ฯ๐ =20 ฯ๐๐ฅ =450
25
26
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 14
๐๐๐๐, าง๐ฅ =ฯ๐๐ฅ
ฯ๐=450
20= 22.5
๐๐๐๐๐๐,๐ = ๐ฟ๐ +
๐2โ ๐น
๐๐๐ถ = 19.5 +
202โ 5
85 = 22.63
๐๐๐๐ = ๐ฟ +๐1
๐1+ ๐2๐ถ = 19.5 +
5
5 + 35 = 22.63
1.2.3 Calculate median and mode for grouped data by using graph
Calculate median by using an ogive
Where N is the total frequency of the distribution.
27
28
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 15
Example 12:
Find the median for the data below by using formula and an ogive.
Marks Frequency Cumulative
Frequency
20-29 2 2
30-39 4 6
40-49 8 14
50-59 12 26
60-69 6 32
70-79 4 36
Solution:
Marks Frequency Cumulative Frequency Upper boundary
10-19 0 0 19.5
20-29 2 2 29.5
30-39 4 6 39.5
40-49 8 14 49.5
50-59 12 26 59.5
60-69 6 32 69.5
70-79 4 36 79.5
29
30
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 16
By using ogive:
5
30
25
20
15
10
28.5 29.5 39.5 49.5 59.5 69.5
Marks
Cu
mu
lative
fre
qu
en
cy
79.5
35
40
Median=52.5
๐ฟ๐๐๐๐ก๐๐๐ ๐๐ ๐๐๐๐๐๐ =๐
2=36
2= ๐๐
๐น๐๐๐ ๐๐ ๐๐๐๐ฃ๐,๐๐๐๐๐๐ = ๐๐.๐
Calculate mode by using Histogram :
Steps:
1. Identify the bar representing the mode class.
2. Joint the top vertices of the bar to the top vertices of the adjacent bars.
3. Read the value on the horizontal axis of the point of intersection of
the lines obtained.
31
32
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 17
Example 13:
The data below shows test marks of a group of students. Find the Mode for
the data below by using formula and histogram.
Test
Marks
No. of
students
50 โ 59 4
60 โ 69 8
70 โ 79 18
80 โ 89 16
90 โ 99 2
33
34
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 18
Solution:
Test Marks No. of students Class boundaries
50 โ 59 4 49.5 โ 59.5
60 โ 69 8 59.5 โ 69.5
70 โ 79 18 69.5 โ 79.5
80 โ 89 16 79.5 โ 89.5
90 โ 99 2 89.5 โ 99.5
Title: Studentโs Test Score
2
18
16
14
12
10
8
6
4
49.5 59.5 69.5 79.5 89.5 99.5Test Marks
Nu
mb
er
of st
ud
en
ts
Mode= 77.5
35
36
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 19
TUTORIAL EXERCISE 1.
1.2.4 Calculate mean deviation, variance, standard deviation for
ungrouped and grouped data
Data can be "distributed" (spread out) in different ways.
37
38
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 20
But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this:
39
40
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 21
The Standard Deviation is a measure of how spread out numbers
are.
Example: 95% of students at school are between 1.1m and 1.7m tall.
Assuming this data is normally distributed can you calculate the mean and standard deviation?
The mean is halfway between 1.1m and 1.7m:
Mean = (1.1m + 1.7m) / 2 = 1.4m
95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:
1 standard deviation = (1.7m-1.1m) / 4
= 0.6m / 4 = 0.15m
41
42
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 22
And this is the result:
43
44
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 23
1.2.4 Calculate mean deviation, variance, standard deviation for
ungrouped and grouped data
Mean deviation of ungrouped data
Example 14:
Determine mean deviation from the data below:
Solution :
45
46
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 24
Mean deviation, ๐ฅ ๐ โ เดฅ๐
23 2.5
20 0.5
15 5.5
17 3.5
18 2.5
30 9.5
ฯ ๐ โ เดฅ๐ =24
Example 15:
Determine mean deviation from the data below:
47
48
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 25
๐ฅ ๐ ๐๐ฅ ๐ โ เดฅ๐ ๐ โ เดฅ๐ ๐
3 8 24 1.78 14.24
4 3 12 0.78 2.34
5 7 35 0.22 1.54
6 5 30 1.22 6.10
7 4 28 2.22 8.88
๐ = ๐๐ ๐๐ = ๐๐๐ ฯ ๐ โ เดฅ๐ ๐ =33.10
Solution :
Mean, Mean deviation,
Mean Deviation of grouped data
49
50
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 26
Example 16: Determine the mean deviation of the data below
๐ช๐๐๐๐ ๐๐๐๐๐๐๐๐ ๐
25 โ 35 5
36 โ 46 8
47 โ 57 14
58 โ 68 11
69 โ 79 7
80 โ 90 5
๐ช๐๐๐๐ ๐๐๐๐๐๐๐๐ ๐ ๐ ๐๐ ๐โ เดฅ๐ ๐ โ เดฅ๐ ๐
25 โ 35 5 30 150 26.84 134.20
36 โ 46 8 41 328 15.84 126.72
47 โ 57 14 52 728 4.84 67.76
58 โ 68 11 63 693 6.16 67.76
69 โ 79 7 74 518 17.16 120.12
80 โ 90 5 85 425 28.16 140.80
50 2842 657.36
Solution :
Mean, Mean deviation,
51
52
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 27
Variance
Variance measures how much the values set of data
vary from the mean of the set of data. The larger
value of variance indicates a greater dispersion of
the values in a set of data from its mean.
Variance
Ungrouped Data (raw data),
Ungrouped Data (FDT) & Grouped Data,
๐ 2 =ฯ ๐ฅ โ าง๐ฅ 2
๐;๐คโ๐๐๐ าง๐ฅ =
ฯ๐ฅ
๐๐ฅ=dataาง๐ฅ=mean of data
n=number of values of the data
๐ 2 =ฯ ๐ฅ โ าง๐ฅ 2๐
ฯ๐
๐ฅ=midpointาง๐ฅ=mean of dataฯ๐ =sum of frequency
๐ 2 =ฯ๐๐ฅ2
ฯ๐โ
ฯ๐๐ฅ
ฯ๐
2
53
54
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 28
Standard Deviation
Standard deviation is also a measure which tells us
how much values in a set of data disperse from the
mean of the set of data.
@ ๐ = ๐ 2
Example 17:
Determine the variance and standard deviation of each of the following
3 5 8 2 4 11 and 9
Solution :
55
56
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 29
๐ 2 =ฯ ๐ฅ โ าง๐ฅ 2
๐
=68
7
= 9.71
๐ = ๐ 2
= 9.71
= 3.12
๐ฅ ๐ฅ โ าง๐ฅ ๐ฅ โ าง๐ฅ 2
3 3 9
5 1 1
8 2 4
2 4 16
4 2 4
11 5 25
9 3 9
68
Example 18:
Determine the variance and standard deviation of each of the following
Solution :าง๐ฅ =
ฯ๐ฅ
๐
=0๐ฅ3 + 1๐ฅ4 + 2๐ฅ6 + 3๐ฅ5 + (4๐ฅ2)
3 + 4 + 6 + 5 + 2
=39
20
= 1.95
57
58
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 30
๐ฅ ๐ ๐๐ฅ ๐ฅ โ าง๐ฅ 2 ๐ฅ โ าง๐ฅ 2๐
0 3 0 3.80 11.4
1 4 4 0.90 4.9
2 6 12 0.0025 0.015
3 5 15 1.10 5.5
4 2 8 4.20 8.4
20 39 30.22
=30.22
20
=1.511
๐ = ๐ 2
= 1.511
=1.23
๐ 2 =ฯ ๐ฅ โ าง๐ฅ 2๐
ฯ๐
Example 19: Determine the variance and standard deviation of :
Class interval f
25 - 35 536 - 46 847 - 57 1458 - 68 1169 - 79 780 - 90 5
59
60
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 31
Solution :๐ช๐๐๐๐ ๐๐๐๐๐๐๐๐ ๐ ๐ ๐๐ ๐ฅ โ าง๐ฅ ๐ฅ โ าง๐ฅ 2๐
25 โ 35 5 30 150 26.84 3601.93
36 โ 46 8 41 328 15.84 2007.24
47 โ 57 14 52 728 4.84 327.96
58 โ 68 11 63 693 6.16 417.40
69 โ 79 7 74 518 17.16 2061.26
80 โ 90 5 85 425 28.16 3964.93
50 2842 12,380.72
๐ 2 =ฯ ๐ฅ โ าง๐ฅ 2๐
ฯ๐
Variance, Standard deviation,
๐ = ๐ 2 = 247.61 = ๐๐.๐๐=12380.72
50= ๐๐๐.๐๐
Solution :๐ช๐๐๐๐ ๐๐๐๐๐๐๐๐ ๐ ๐ ๐๐ ๐๐ ๐๐๐
25 โ 35 5 30 150 900 4500
36 โ 46 8 41 328 1681 13448
47 โ 57 14 52 728 2704 37856
58 โ 68 11 63 693 3969 43659
69 โ 79 7 74 518 5476 38332
80 โ 90 5 85 425 7225 36125
50 2842 173920
Variance, Standard deviation,
๐ = ๐ 2 = 247.61 = ๐๐.๐๐=173920
50โ
2842
50
2
= ๐๐๐.๐๐
๐ 2 =ฯ๐๐ฅ2
ฯ๐โ
ฯ๐๐ฅ
ฯ๐
2
OR
61
62
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 32
TUTORIAL EXERCISE:
TUTORIAL EXERCISE 2
63
64
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 33
1.2.5 Calculate quartiles, deciles and percentiles for grouped
data by using formula and graph
Quartile: Divide a set of data into
four equal parts with all the data
arranged in ascending or
descending order.
Q1 = first quartile
Q2 = second quartile = median
Q3 = third quartile
Interquartile range = Q3 - Q1
๐ช =class size
๐ธ๐ = ๐ณ๐ธ๐ +
๐๐ต4 โ ๐ญ
๐๐ธ๐๐ช; ๐ = 1, 2, 3
๐ณ๐ธ๐ = lower boundary of the class in which the first quartile lies
๐ต =sum of frequency
๐ญ =cumulative frequency before the class in which first quartile lies
๐๐ธ๐ =frequency of the class in which the first quartile lies
Calculate quartiles, deciles and percentiles for grouped
data by using formula
65
66
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 34
Quartile of ungrouped data
Find first quartile, third quartile and interquartile range for data below:
Example 20:
Example 21:
67
68
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 35
Decile: Any one of the numbers or values in a series dividing the distribution
of the individuals in the series into ten groups of.
๐ซ๐ = ๐ณ๐ซ๐+
๐๐ต10
โ ๐ญ
๐๐ซ๐
๐ช; ๐ = 1, 2, 3,โฆ 9
๐ช = class size
๐ณ๐ซ๐= lower boundary of the class in which the decile lies
๐ต = sum of frequency
๐ญ = cumulative frequency before the class in which decile lies
๐๐ซ๐= frequency of the class in which the decile lies
Calculate quartiles, deciles and percentiles for grouped
data by using formula
Percentile: Any one of the numbers or values in a series dividing the distribution
of the individuals in the series into hundred groups of.
๐ท๐ = ๐ณ๐ท๐ +
๐๐ต100
โ ๐ญ
๐๐ท๐๐ช; ๐ = 1, 2, 3,โฆ 99
๐ณ๐ท๐ = lower boundary of the class in which the percentile lies
๐ต = sum of frequency
๐ญ = cumulative frequency before the class in which percentile lies
๐๐ท๐ = frequency of the class in which the percentile lies
๐ช = class size
Calculate quartiles, deciles and percentiles for grouped
data by using formula
69
70
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 36
Example 22 :The following table shows the marks of a group of students.
Marks Frequency
60-62 6
63-65 18
66-68 40
69-71 28
72-74 8
Using formula, find the:
(a) First quartile, Q1
(b) Third quartile, Q3
(c) Median
(d) Seventh decile, D7
(e) 85th percentile, P85
Calculate quartiles, deciles and percentiles for grouped
data by using formula
Solution:Marks Frequency (f) Cumulative
Frequency
Class
boundaries
60-62 6 6 59.5 โ 62.5
63-65 18 24 62.5 โ 65.5
66-68 40 64 65.5 โ 68.5
69-71 28 92 68.5 โ 71.5
72-74 8 100 71.5 โ 74.5
71
72
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 37
(a) First quartile, Q1
๐1 = ๐ฟ๐1 +
๐4โ ๐น
๐๐1๐ถ
= 65.5 +
1004
โ 24
403
= ๐๐.๐๐
(b) Third quartile, Q3
๐3 = ๐ฟ๐3 +
3๐4โ ๐น
๐๐3๐ถ
= 68.5 +
3)(1004 โ 64
283
= ๐๐.๐๐
= ๐๐.๐๐
(c) Median
๐2 = ๐ฟ๐2 +
2๐4 โ ๐น
๐๐2๐ถ
= 65.5 +
(2)(100)4 โ 24
403
= ๐๐.๐๐
(d) First decile, D1
๐ท1 = ๐ฟ๐ท1 +
1๐10 โ ๐น
๐๐ท1๐ถ
= 62.5 +
1(100)10 โ 6
183
73
74
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 38
= ๐๐.๐๐
(e) 95th percentile, P95
๐95 = ๐ฟ๐๐ +
๐๐100
โ ๐น
๐๐๐๐ถ
= 71.5 +
95(100)100 โ 92
83
Example 23: The following table shows the marks of a group of students.
Marks Frequency
60-62 6
63-65 18
66-68 40
69-71 28
72-74 8
Draw an ogive and find the:
(a) First quartile, Q1
(b) Third quartile, Q3
(c) Seventh decile, D1
(d) 95th percentile, P95
Calculate quartiles, deciles and percentiles for grouped
data by using graph
75
76
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 39
๐ Quartile = the ๐
4๐กโ value of the cumulative frequency, where ๐ = 1, 2, 3
By using an ogive,
Interquartile range= 3๐๐ ๐๐ข๐๐๐ก๐๐๐ โ 1๐ ๐ก ๐๐ข๐๐๐ก๐๐๐
Decile = ๐
10๐กโ value of the cumulative frequency, where, ๐ = 1, 2, 3,โฆ 9
Percentile = ๐
100๐กโ value of the cumulative frequency, where, ๐ = 1, 2, 3, โฆ99
Solution:
Marks Frequency (f) Cumulative
Frequency
Class
boundaries
57-59 0 0 56.5 โ 59.5
60-62 6 6 59.5 โ 62.5
63-65 18 24 62.5 โ 65.5
66-68 40 64 65.5 โ 68.5
69-71 28 92 68.5 โ 71.5
72-74 8 100 71.5 โ 74.5
77
78
.
https://cikguamirul.wordpress.com/electrical-engineering-mathematics/ 40
(d) Location on the y-axis:1
10ร 100 = 10;
From the graph, ๐ท1 =63.4
(b) Location on the y-axis:3
4ร 100 = 75;
From the graph, ๐3 =69.5
D1=63.4Q3=69.5 P95=72.1
Median=67.3
Q1=65.5
(a) Location on the y-axis: 1
4ร 100 = 25;
From the graph, ๐1 =65.5
(c) Location on the y-axis:1
2ร 100 = 50;
From the graph, ๐2=67.3
(d) Location on the y-axis: 95
100ร 100 = 95;
From the graph, ๐95 =72.1
TUTORIAL EXERCISE 3.1:
12 10 22 23 25 41 41 20 90 25
65 13 89 47 33 52 47 65 66 32
55 13 88 37 81 53 50 64 71 90
45 90 19 57 73 53 11 30 72 44
34 87 17 67 80 11 14 15 70 40
i. Based on the data above, draw a โless thanโ ogive graph, using 5-14 as a first class
ii. Then, determine the first quartile, 7th decile, and median from the given data.
79
80