yte

2. Ungrouped Data
 Measures Of Dispersion or Variation
Grouped data
 Measures of Central Tendency
 Measures of central/non central position

3. MEASURE OF DISPERSION OR VARIATION
- indicate how the observations are scattered or
dispersed about the average.
1.Range
2.Variance
3.Standard Deviation
4.Coefficient of Variation

4. Range
– the difference between the highest value
and the smallest value in the data set.
R = (Highest value – Lowest value)

5. Variance
- Is the average of the squared deviation from the
mean.
𝜎2 =
𝑖=1
𝑁
(𝑥𝑖 − 𝜇)2
𝑁
𝑓𝑜𝑟 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑠2 =
𝑖=1
𝑛
(𝑥𝑖 − 𝑥)2
𝑛 − 1
𝑓𝑜𝑟 𝑠𝑎𝑚𝑝𝑙𝑒

6. Standard Deviation
- Is the positive square root of the average of
the squared deviation from the mean.
For Population: 𝜎 = 𝜎2
For Sample : 𝑠 = 𝑠2

7. Coefficient of Variation
- Is the ratio of the standard deviation to the
mean.
For Population: 𝐶𝑉 =
𝜎
𝜇
× 100%
For Sample : 𝐶𝑉 =
𝑆
𝑋
× 100%

9. MEASURES OF CENTRL TENDENCY FOR
GROUPED DATA
 Arithmetic Mean
𝜇 = 𝑖=1
𝑘
𝑓 𝑖 𝑥 𝑖
𝑖=1
𝑘
𝑓 𝑖
for population
𝑥 = 𝑖=1
𝑘
𝑓 𝑖 𝑥 𝑖
𝑖=1
𝑘 𝑓 𝑖
for sample
- The mean of a frequency distribution table is simply a weighted
mean of the class marks where frequencies partake the nature of
the weights.

10. MEASURES OF CENTRL TENDENCY FOR GROUPED DATA
 Median ( 𝜇, 𝑥 )
𝜇 = 𝐿𝐶𝐵 𝑚 + 𝑐 [
𝑛
2
− 𝐹 𝑚−1
𝑓𝑚
]
𝑥 =𝐿𝐶𝐵 𝑚 + 𝑐 [
𝑛
2
−𝐹 𝑚−1
𝑓 𝑚
]
Where :
𝐿𝐶𝐵 𝑚 = is the lower class boundary of the median class
𝑐 = class width
𝐹 𝑚−1 = cumulative frequency before the cumulative frequency of the
median class.
𝑓𝑚 = frequency of the median class
n = number of observation

11. MEASURES OF CENTRL TENDENCY FOR GROUPED DATA
 Mode ( 𝜇, 𝑥 )
𝜇 = 𝐿𝐶𝐵 𝑚𝑜 + 𝑐 [
𝑓 𝑚0 − 𝑓1
2𝑓𝑚𝑜 − 𝑓1 − 𝑓2
]
𝑥 = 𝐿𝐶𝐵 𝑚𝑜 + 𝑐 [
𝑓 𝑚0 − 𝑓1
2𝑓𝑚𝑜 − 𝑓1 − 𝑓2
]
where
𝐿𝐶𝐵 𝑚𝑜= is the lower class boundary of the
modal class
c = the class width
𝑓 𝑚0 = frequency of the modal class
𝑓1 = frequency before the modal class
𝑓2 = frequency after the modal class.

12. VARIANCE
𝜎2=
𝑁 𝑖=1
𝑘
𝑓𝑖 𝑥𝑖
2
− ( 𝑖=1
𝑘
𝑓𝑖 𝑥𝑖 )2
𝑁2
for population
𝑠2
=
𝑛 𝑖=1
𝑘
𝑓𝑖 𝑥𝑖
2
− ( 𝑖=1
𝑘
𝑓𝑖 𝑥𝑖 )2
𝑛(𝑛 − 1)
𝑓𝑜𝑟 𝑠𝑎𝑚𝑝𝑙𝑒

13. Measure of Location
 PERCENTILE
 DECILE
 QUARTILE

14. PERCENTILE-
- Divides the distribution into 100 equal
parts.
- Denoted by 𝑃1, 𝑃2, 𝑃3, 𝑃4, 𝑃5,…….., 𝑃99
𝑃𝑖 = 𝐿𝐶𝐵𝑖 + 𝑐 [
𝑖𝑛
100
− 𝐹𝑖−1
𝑓𝑖
]

15. DECILE
- Divides the distribution into ten parts…
- Denoted by 𝐷1, 𝐷2, 𝐷3, 𝐷4, 𝐷5, 𝐷6, 𝐷7, 𝐷8,
𝐷9
Note that:
𝐷1= 𝑃10 𝐷6= 𝑃60
𝐷2=𝑃20 𝐷7= 𝑃70
𝐷3=𝑃30 𝐷8= 𝑃80
𝐷4=𝑃40 𝐷9= 𝑃90
𝐷5=𝑃50

16. QUARTILE (𝑄𝑖)
- Divides the distribution into four parts.
- Denoted by 𝑄1, 𝑄2, 𝑄3
Note that :
𝑄1 = 𝑃25
𝑄2 = 𝑃50
𝑄3 = 𝑃75