The median value for grouped frequency distributions

The median value for grouped frequency distributions is given by the
formula:
˜
X = L
˜
X
+


n
2
- F
˜
X - 1
f
˜
X


× w
where
n =
k
i =1
f
i
is the total number of observations,
f
˜
70 / 159
Introduction to Statistics
Ch-3: Measures of Central Tendency
Median and Quantiles
Median - Contd.
Example: Find the median mark of the students score data and interpret
it.
Introduction to Statistics
Ch-3: Measures of Central Tendency
Median and Quantiles
Solutions - Contd.
71 / 159 72 / 159

Introduction to Statistics
Ch-3: Measures of Central Tendency
Mode
Mode
Mode
It is a value of a particular type of items which occur most frequently.
Ungrouped (individual series) cases: The value appearing most
frequently (the most frequent value) is taken as the modal value.
Examples: Find the mode of the following data sets.
1. 110, 113, 116, 116, 118, 118, 118, 121 and 123.
2. 2, 3, 5, 7 and 8.
3. 15, 18, 18, 18, 20, 22, 24, 24, 24, 26 and 26.
4. 5, 6, 6, 7, 9, 9, 10, 12 and 12.
5. 1, 1, 0, 1, 0, 0, 0, 2, 4 and 3.
73 / 159
Introduction to Statistics
Ch-3: Measures of Central Tendency
Mode
Solutions
1. Since 118 occurs more than other values, the mode is 118.
2. Each value occurs once (equally frequent), the data has no mode.
3. 18 and 24 occur three times, hence the modal values are 18 and 24
(bi-modal).
4. Tri-modal (multi-modal): 6, 9 and 12.
5. The modal value here is 0 as it occurs more number of times than other
values.
74 / 159
Introduction to Statistics
Ch-3: Measures of Central Tendency
Mode
Mode - Contd.
Grouped (continuous) series : In a grouped frequency distribution, the
modal value is located in the class with highest frequency and that class
is the modal class.
ˆ
X = L
ˆ
X
+
f
ˆ
X
- f
ˆ
X - 1
( f
ˆ
X
- f
ˆ
X - 1
) + ( f
ˆ
X
- f
ˆ
X +1
)
× w
Example: Find the modal score of the students score data.
The class having highest frequency is ⇒ 26 . 5 - 30 . 5, hence it is the modal
class.
ˆ
X = 26 . 5 +
12 - 10
(12 - 10) + (12 - 7)
× 4 = 26 . 5 + 1 . 14 = 27 . 64
75 / 159
Introduction to Statistics
Ch-4: Measures of Variation
Definition of Variation
What is Variation?
Example: Find the mean and median prices of the four cities and
interpret it.
A 30 30 30 30 30
B 28 29 30 31 32
C 10 15 30 45 50
D 0 5 30 55 60
Are all the four data sets the same?
If yes, why?
If no, why not?
Variation
Variation (dispersion) may be defined as the extent of scatteredness of value
around the measures of central tendency.
76 / 159

Introduction to Statistics
Ch-4: Measures of Variation
Definition of Variation
Objectives of Measures of Variation
1. To have an idea about the reliability of the measures of central
tendency . If the degree of scatterdness is large, an average is less
reliable.