This document discusses measures of central tendency and dispersion for grouped and ungrouped data. It defines range, variance, standard deviation, and coefficient of variation as measures of dispersion that indicate how observations are scattered around the average. It also provides the formulas to calculate the mean, median, and mode for grouped data as measures of central tendency, and the lower class boundary method to determine median and mode for grouped data.

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%

8. 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

9. 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

10. 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.