STA 113 2.0 Descriptive Statistics

Measures of Central Tendency and Variability: Grouped Data

Dr. Thiyanga S. Talagala
Department of Statistics, Faculty of Applied Sciences
University of Sri Jayewardenepura, Sri Lanka

Data

Class interval	Frequency (f)
(1, 3]	16
(3, 5]	2
(5, 7]	4
(7, 9]	3
(9, 11]	9
(11, 13]	6

Data

Class interval	Frequency (f)	Class midpoint (M)
(1, 3]	16	2
(3, 5]	2	4
(5, 7]	4	6
(7, 9]	3	8
(9, 11]	9	10
(11, 13]	6	12

Mean

\[Mean = \frac{\sum fM}{\sum f}\]

Mode

In-class demo

Plot the histogram.

Median

\[Median = L_M + [\frac{\frac{(\sum f)}{2}-F}{f_M}]W\]

\(L_M\) - lower limit of the class containing the median

\(\sum f\) - total number of observations

\(F\) - the cumulative frequency up to the lower limit of the class containing the median

\(f_M\) - frequency of the class containing the median

\(W\) - width of the class containing the median

Quartiles

\[Q_1 = L_{q1} + [\frac{\frac{(\sum f)}{4}-F_{q1}}{f_{q1}}]W\]

\(L_{q1}\) - lower limit of the class containing the first quartile

\(\sum f\) - total number of observations

\(F_{q1}\) - the cumulative frequency up to the lower limit of the class containing the first quartile

\(f_M\) - frequency of the class containing the median

\(W\) - width of the class containing the median

Your turn: Write the formula for Q3

Population variance

\[\sigma^2 = \frac{\sum f(M-\mu)^2}{\sum f}\]

\(M\) - Class midpoint

Sample variance

\[s^2 = \frac{\sum f(M-\bar{x})^2}{\sum f -1}\]

\(M\) - Class midpoint