This document discusses various statistical methods for analyzing data distributions, including frequency/percentage distributions, weighted means, measures of central tendency (median), measures of position (deciles and percentiles), and quartiles. Frequency/percentage distributions are used to determine the percentage of respondents for each item. Weighted means are used to calculate the average value. Measures of position like deciles and percentiles divide the data set into equal parts to identify cutoff values. Quartiles divide the data into four equal parts. Examples are provided for calculating deciles, percentiles, and quartiles for different data sets.

2. Statistical Treatment

3. Frequency and Percentage Distribution
Used to determine the percentage usually for data on
profile (e.g. level, age, gender, etc.)
% =
𝑓
𝑁
𝑥 100
where:
% - percentage
f – frequency or number of
respondents for every item
N - total no. of respondents
P =
𝑓
𝑁
𝑥 100%
where:
P - percentage
f – frequency or number of
respondents for every item
N - total no. of respondents

7. Weighted mean
Used to get the average or central value (e.g. level,
extent, status, etc.)
𝒙 =
𝒇𝒙
𝒏
where:
𝒙 - weighted mean
f – the frequency of scores
n - number of respondents

14. Measures of Position
Ungrouped data
Also known as Measures of Location. These are
identified as values which divides the data set into a
desired equal parts.
4 parts – Quartile
10 parts – Decile
100 parts - Percentile

15. Decile
These numbers divide the data set into 10 equal parts.
𝐷𝑥 =
𝑥
10
(𝑛 + 1)
𝐷2 =
2
10
𝑛 + 1
𝐷2 =
1
5
(𝑛 + 1)
𝐷5 =
5
10
𝑛 + 1
D5 =
1
2
(𝑛 + 1)
𝑚𝑒𝑑𝑖𝑎𝑛
𝐷7 =
7
10
𝑛 + 1

16. Data: 5, 6, 7, 7, 10, 11, 12, 13, 14,
17, 18, 18, 19, 22, 25, 28, 33, 34, 34,
35, 37, 39, 42, 45, 47, 48, 49, 52, 54
• 1st decile=7
• 2nd decile=11
• 3rd decile=14
• 4th decile=18
• 5th decile=25
• 6th decile=34
• 7th decile=37
• 8th decile=45
• 9th decile=49
𝐷1 =
1
10
29 + 1
D1 = 3𝑟𝑑 = 7
𝐷2 =
2
10
29 + 1
D2 = 6𝑡ℎ = 11
𝐷3 =
3
10
29 + 1
D3 = 9𝑡ℎ = 14

17. Data: 5, 6, 7, 7, 10, 11, 12, 13, 14, 17,
18, 18, 19, 22, 25, 28, 33, 34, 34, 35, 37,
39, 42, 45, 47, 48, 49, 52, 54, 55, 62, 65
• 2nd decile
• 3rd decile
• 5th decile
• 7th decile
• 8th decile
𝐷2 =
2
10
32 + 1
D2 =
1
5
33
𝐷2 = 6.6
6𝑡ℎ 𝑡𝑒𝑟𝑚 + 0.6 7𝑡ℎ 𝑡𝑒𝑟𝑚 − 6𝑡ℎ 𝑡𝑒𝑟𝑚
11 + 0.6 12 − 11
11 + 0.6 1
𝟏𝟏. 𝟔

18. Data: 5, 6, 7, 7, 10, 11, 12, 13, 14, 17,
18, 18, 19, 22, 25, 28, 33, 34, 34, 35, 37,
39, 42, 45, 47, 48, 49, 52, 54, 55, 62, 65
• 2nd decile
• 3rd decile
• 5th decile=30.5
• 7th decile=42.3
• 8th decile=48.4
𝐷3 =
3
10
32 + 1
D2 = 9.9𝑡ℎ
9𝑡ℎ 𝑡𝑒𝑟𝑚 + 0.9 10𝑡ℎ 𝑡𝑒𝑟𝑚 − 9𝑡ℎ 𝑡𝑒𝑟𝑚
14 + 0.9 17 − 14
14 + 0.9 3
14 + 2.7
3𝑟𝑑 𝑑𝑒𝑐𝑖𝑙𝑒 = 𝟏𝟔. 𝟕

19. Percentile
These numbers divide the data set into 100
equal parts.
𝐷𝑥 =
𝑥
100
(𝑛 + 1)

20. Data: 5, 6, 7, 7, 10, 11, 12, 13, 14, 17,
18, 18, 19, 22, 25, 28, 33, 34, 34, 35, 37,
39, 42, 45, 47, 48, 49, 52, 54, 55, 62, 65
• 22nd percentile
• 43rd percentile
• 50th percentile
• 51st percentile
• 67th percentile
• 84th percentile
𝑃22 =
22
100
32 + 1
𝑃22 =
11
50
33
𝑃22 = 7.26
7𝑡ℎ 𝑡𝑒𝑟𝑚 + 0.26 8𝑡ℎ 𝑡𝑒𝑟𝑚 − 7𝑡ℎ 𝑡𝑒𝑟𝑚
12 + 0.26 14 − 13
12 + 0.26 1
𝟏𝟐. 𝟐𝟔

21. Special Similarities
Measures of Position is directly connected
with Measures of Central Tendency.
The 5th decile, the 50th percentile are equal
to the median.

23. Quartile
These numbers divide the data set into 4
equal parts.
UNGROUPED DATA

24. Quartile
𝑄𝑥 =
𝑥
4
(𝑛 + 1)
𝑪𝒂𝒔𝒆 𝟏.
N is an odd number.
𝑁 + 1 is divisible by 4.
67, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 87, 89, 91
𝑄1 =
1
4
15 + 1
𝑄1 = 4𝑡ℎ = 72
𝑄2 =
1
2
15 + 1
𝑄2 = 8𝑡ℎ = 79
𝑄3 =
3
4
15 + 1
𝑄3 = 12𝑡ℎ = 84

25. Quartile
𝑄𝑥 =
𝑥
4
(𝑛 + 1)
𝑪𝒂𝒔𝒆 𝟐.
N is an odd number.
𝑁 + 1 is not divisible by 4.
67, 69, 71, 72, 74, 75, 77, 78, 79, 80, 82, 83, 84, 87, 89, 91, 92
𝑄1 =
1
4
17 + 1
𝑄1 = 4.5𝑡ℎ
4𝑡ℎ + 5𝑡ℎ
2
72 + 74
2
𝟕𝟑
𝟕𝟑
𝑄2 =
1
2
17 + 1
𝑄2 = 9𝑡ℎ = 79
𝑄3 =
3
4
17 + 1
𝑄3 = 13.5𝑡ℎ
13𝑡ℎ + 14𝑡ℎ
2
84 + 87
2
𝟖𝟓. 𝟓
𝟖𝟓. 𝟓

26. Quartile
Average of the
𝑥𝑁
4
𝑡ℎ and
𝑥𝑁
4
+ 1 𝑡ℎ term
𝑪𝒂𝒔𝒆 𝟑.
N is an even number.
𝑁 is divisible by 4.
67, 69, 70, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 87, 89, 91
𝑄1
1
4
16
4𝑡ℎ
71
𝟕𝟏. 𝟓
1
4
16 + 1
5𝑡ℎ
72
71 + 72
2
71.5
𝑄2
1
2
16
8𝑡ℎ
77
1
4
16 + 1
5𝑡ℎ
79
77 + 79
2
78
𝟕𝟖
Average of the
𝑥
4
(𝑁) and
𝑥
4
𝑁 + 1

27. Quartile
Average of the
𝑥𝑁
4
𝑡ℎ and
𝑥𝑁
4
+ 1 𝑡ℎ term
𝑪𝒂𝒔𝒆 𝟑.
N is an even number.
𝑁 is divisible by 4.
67, 69, 70, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 87, 89, 91
𝑄3
3
4
16
12𝑡ℎ
83
𝟖𝟑. 𝟓
3
4
16 + 1
13𝑡ℎ
84
83 + 84
2
𝟖𝟑. 𝟓
Average of the
𝑥
4
(𝑁) and
𝑥
4
𝑁 + 1

28. Quartile
𝑪𝒂𝒔𝒆 𝟒.
N is an even number.
𝑁 is not divisible by 4.
Average of the
1
2
𝑁 𝑡ℎ and
1
2
𝑁 + 1 𝑡ℎ term
Second Quartile
𝑥
4
𝑁 + 0.5 𝑡ℎ term
First and Third Quartile
67, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 85, 87, 89, 91, 93, 95

29. Quartile
𝑪𝒂𝒔𝒆 𝟒.
N is an even number.
𝑁 is not divisible by 4. 𝑥
4
𝑁 + 0.5 𝑡ℎ term
First and Third Quartile
67, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 85, 87, 89, 91, 93, 95
𝑄1
1
4
18 + 0.5
5𝑡ℎ
74
𝑄3
3
4
18 + 0.5
14𝑡ℎ
87

30. Quartile
𝑪𝒂𝒔𝒆 𝟒.
N is an even number.
𝑁 is not divisible by 4.
Average of the
1
2
𝑁 𝑡ℎ and
1
2
𝑁 + 1 𝑡ℎ term
Second Quartile
67, 69, 71, 72, 74, 75, 77, 79, 80, 82, 83, 84, 85, 87, 89, 91, 93, 95
𝑄1
1
4
18 + 0.5
5𝑡ℎ
74
𝑄3
3
4
18 + 0.5
14𝑡ℎ
87
𝑄2
1
2
18
9𝑡ℎ
80
𝟖𝟏
1
2
18 + 1
10𝑡ℎ
82
80 + 82
2
81

32. Quartile
𝑄𝑥 =
𝑥
4
(𝑛 + 1)
𝑪𝒂𝒔𝒆 𝟏.
N is an odd number.
𝑁 + 1 is divisible by 4.
𝑪𝒂𝒔𝒆 𝟐.
N is an odd number.
𝑁 + 1 is not divisible by 4.

33. Quartile
𝑪𝒂𝒔𝒆 𝟑.
N is an even number.
𝑁 is divisible by 4.
Average of the
𝑥
4
(𝑁) and
𝑥
4
𝑁 + 1
𝑪𝒂𝒔𝒆 𝟒.
N is an even number.
𝑁 is not divisible by 4.
Average of the
1
2
𝑁 𝑡ℎ and
1
2
𝑁 + 1 𝑡ℎ term
Second Quartile
𝑥
4
𝑁 + 0.5 𝑡ℎ term
First and Third Quartile