This document discusses various statistical measures used to summarize and describe data distributions, including measures of central tendency (mean, median, mode), measures of dispersion (range, interquartile range, standard deviation), and measures of relative position (percentiles, quartiles, z-scores). It provides definitions, formulas, and examples for calculating each measure and interpreting the results.

1. CHAPTER 4- Statistics
a. Measures of Central Tendency
b. Measures of Dispersion
c. Measures of Relative Position

2. Measures of Central Tendency
The term central tendency refers to the middle, or
typical, value of a set of data, which is most
commonly measured by using the three m's: mean,
median, and mode. The mean, median, and mode
are known as the measures of central tendency.
2

3. Today we will understand:
• Measures of Central Tendency
* Mean
* Median
* Mode
3

4. 4

5. Mean
● It is the most commonly used measure of
the center of data.
● It is also referred as the arithmetic average
5

6. Example of Mean
6

7. 7

8. ✖ The Population mean is represented
by the Greek letter – μ.
✖ The Sample mean is represented by
the Greek letter – x.
8

9. The mean is calculated by using
the equations
9

10. 10

11. 11

12. 12

13. 13

14. 14

15. 15

16. What is the Mode in statistics
✖ The Mode is the least used measure of
central tendency.
✖ The Mode of a data set is the
number/observation/value that occurs
most frequently in the set.
16

17. 17

18. 18

19. 19

20. Measures of
Dispersion
20

21. INTRODUCTION
● So far we have looked at ways of summarising
data by showing some sort of average (central
tendency).
● But it is often useful to show how much these
figures differ from the average.
● This measure is called dispersion.
21

22. Measures of dispersion
●These are ways of showing dispersion:
- Range
- Inter-quartile range
- Semi- interquartile range (quartile deviation)
- Coefficient of quratile deviation
- Mean deviation
- Variance
- Coefficient of variation
22

23. The Range
● The range is defined as the difference between
the larges score in the set of data and the
smallest score in the set of data, XL - Xs
● What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
● The largest score (XL) is 9: the smallest score
(Xs) is 1; the range is XL- Xs = 9-1 = 8
23

24. When To Use the Range
● The range is used when
- you have ordinal data or
- you are presenting your results to people with little or no
knowledge of statistics
● The range is rarely used in scientific work as it is fairly
insensitive
- it depends on only two scores in the set of data, XL and Xs
- Two very different sets of data can have the same range:
1 1 1 1 9 vs 1 3 5 7 9
24

25. The Inter-Quartile Range
● The inter-quartile range of the middle half of the
value.
● It is a better measurement to use than the range
because it only refers to the middle half of the
results
● Basically, the extremes are omitted and cannot
affect the answer.
25

26. Example
● To calculate the inter-quartile range we must
first find the quartile.
● These are three quartile, called Q1, Q2, and
Q3. We do not need to worry about Q2 ( this is
the median).
● Q1 is simply the middle value of the bottom half
of the data and Q3 is the middle value of the top
half of data.
26

27. 27

28. Quartile Deviation
● It is the second measure of dispersion, no
doubt improved version over the range. It is
based on the quartile so while calculating this
may require upper quartile (Q3) and lower
quartile (Q1) and then is divided by 2. Hence
it is half of the deference between two
quartiles it is also a semi inter quartile range.
28

29. The formula of Quartile
Deviation is
● ( Q D) = Q3 – Q1
2
29

30. The Semi-Interquartile Range
● The semi-interquartile range ( or SIR)
is defined as the difference of the first
and third quartiles divided by two
- The first quartile is the 25th percentile
- The third quartile is the 75th percentile
● SIR= (Q3-Q1)/2
30

31. Coefficient of Quartile Deviation
● The relative measure of dispersion corresponding to
quartile deviation is known as the coefficient of quartile
deviation.
● QD=Q3-Q1/Q3+Q1
● This will be always less than one and will be positive as
Q3>Q1.
● Smaller value of coefficient of QD indicates lesser
variability.
31

32. Mean Deviation
✖ Mean Deviation is also known as average
deviation. In this case deviation taken
from any average especially Mean,
Median or Mode. While taking deviation
we have to ignore negative items and
consider all of them as positive.
32

33. Mean Deviation
✖
33

34. Standard Deviation
✖ The concept of standard deviation was first
introduced by Karl Pearson in 1893. The
standard deviation is the most useful and
the most popular measure of dispersion.
Just as the arithmetic mean is the most of
all the averages, the standard deviation is
the best of all measures of dispersion.
34

35. Standard Deviation
✖ The standard deviation is one of the most
important measures of dispersion. It is
much accurate than the range or inter
quartile range.
✖ It takes into account all values and is not
unduly affected by extreme values.
35

36. What does it measure
✖ It measures the dispersion ( or spread) of
figures around the mean.
✖ A large number for the standard deviation
means there is a wide spread of values
around the mean, whereas a small number
for the standard deviation implies that the
values are grouped close together around the
mean.
36

37. 37

38. 38

39. 39

40. Example:
✖ We are going to try and find the standard
deviation of the minimum temperatures
of 10 weather stations in Britain on a
winters day.
The temperatures are:
5, 9, 3, 2, 7, 9, 8, 2, 2, 3 ( Centigrade)
40

41. 41

42. 42

43. 43

44. 44

45. 45

46. 46

47. 47

48. 48

49. Measures of
Relative Position
49

50. Contents
50
✖ Percentile
✖ Quartiles
✖ Z- Score.

51. 51
PERCENTILE
✖ A percentile is a measure used in
statistics indicating the value below which
a given percentage of observations in a
group of observations fall.
✖ The formula for percentile is ,

52. 52
✖ The 25th percentile is also called the first
quartile.
✖ The 50th percentile is generally the median.
✖ The 75th percentile is called the third
quartile
✖ The difference between the third and first
quartiles is the interquartile range.

53. 53

54. 54
QUARTILE
✖ Quartiles are the value that divide a list of
numbers into quarters.
✖ Put the list of numbers in order
✖ Then cut the list into four equal parts
✖ The quartiles are at the “cuts “

55. 55
Example
5,7,4,4,6,2,8
● Put them in order: 2,4,4,5,6,7,8
● Cut the list in quarters:
And the result is:
● Quartile 1(Q1)=4
● Quartile 2(Q2), which is also the median ,=5
● Quartile 3(Q3)=7

56. 56
Interquartile Range

57. 57
Box and Whisker Plot

58. 58
Example: Box and Whisker Plot and
Interquartile Range for.

59. 59
In this case all quartiles are between
numbers.
Quartile 1(Q1)=(4+4)/2= 4
Quartile 2(Q2)=(10+11)/2= 10.5
Quartile 3(Q3)=(14+16)/2= 15
Also
The lowest value is 3
The highest value is 18

60. 60
So now we have enough data for the Box and
Whisker Plot
 And the Interquartile Range is:
=> Q3 – Q1= 15-4 = 11

61. 61
STANDARD SCORE or Z- SCORE
 A standard score or Z- score is used when
direct comparison of raw scores is impossible.
 A standard score or Z-score for a value is
obtained by subtracting the mean from the
value and dividing the results by the standard
deviation.

62. 62
 Obtained by subtracting the mean from the value
and dividing the result by the standard deviation.
 The symbol for the standard score is Z.

63. 63
Example:
A national achievement test is administered
annually to 3rd graders. The test has a mean
score of 100 and a standard deviation of 15.
If Jane’s Z- Score is 1.20. What was her
score on the test?

64. 64
Solution
 Z = ( x μ)/ó
 Where Z is the Z-score, X is the value of the
element, μ is the population, and ó is the
standard deviation
 Solving for Jane’s test score (x), we get
 X= ( Z* Ó )= (1.20*15)+100=18+100 = 118

65. Thanks!
Any questions?
65

66. Test 1:
Find the mean,median,mode
1.34,68,84,76,61,61,85,54,44,70,72,83,61
66

67. Test ||:
Interquartile
67