Download App
Google Android
Apple iOS
Huawei
English
English
Malay
Guest
Login
Register
Home
Quiz
Battle
Practice
Class
Classes List
Timetable
Assignments
Learn
Learning Hub
Quick Notes
Videos
Experiments
Textbooks
Login
Register
Download App
Google Android
Apple iOS
Huawei
EN
MS
Learn
Quick Notes
List
Measures of Dispersion
Measures of Dispersion
7.2
Measures of Dispersion
Some formula for a grouped data
\(\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}\)
\(\text{Interquartile range, IQR}=Q_3-Q_1\)
\(\begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned}\)
\(\begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned}\)
\(\begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned}\)
where
\(\begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}\)
Example 5
The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data.
Solution:
\(\begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned}\)
\(\begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned}\)
\(\begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}\)
What is a boxplot?
A
box plot
is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
Example 6
The ogive below shows the masses in g, of 90 starfruits.
a) Construct a box plot based on the ogive.
b) Hence, state the distribution shape of the data.
Solution:
a)
\(\begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned}\)
b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
Measures of Dispersion
7.2
Measures of Dispersion
Some formula for a grouped data
\(\text{range}=\text{midpoint of the highest class} - \text{midpoint of the lowest class}\)
\(\text{Interquartile range, IQR}=Q_3-Q_1\)
\(\begin{aligned} \text{Mean, }\bar{x}=\frac{\sum fx}{\sum f} \end{aligned}\)
\(\begin{aligned} \text{Variance, }\sigma^2=\frac{\sum fx^2}{\sum f}-\bar{x}^2 \end{aligned}\)
\(\begin{aligned} \text{Standard deviation, }\sigma=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}} \end{aligned}\)
where
\(\begin{aligned} &x=\text{midpoint of the class interval}\\ &f=\text{frequency}\\ \end{aligned}\)
Example 5
The frequency table below shows the volumes of water to the nearest litres, used daily by a group of families in a housing area. Calculate the variance and standard deviation of the data.
Solution:
\(\begin{aligned} \text{Mean, }\bar{x}&=\frac{\sum fx}{\sum f} \\ &=\frac{17417.5}{95}\\ &=183.34 \end{aligned}\)
\(\begin{aligned} \text{Variance, }\sigma^2&=\frac{\sum fx^2}{\sum f}-\bar{x}\\ &=\frac{3 215 133.75}{95}-\bigg(\frac{17 417.5}{95}\bigg)^2\\ &=229.1856 \end{aligned}\)
\(\begin{aligned} \text{Standard deviation, }\sigma&=\sqrt{\frac{\sum fx^2}{\sum f}-\bar{x}}\\ &=\sqrt{229.1856} \\ &=15.1389 \end{aligned}\)
What is a boxplot?
A
box plot
is a method to display a group of numerical data graphically based on the five number summary of data. They are the minimum value, first quartile, median, third quartile and maximum value. Similar to the histrogram and frequency polygon, the shape of a distribution can also be identified through the box plot.
Example 6
The ogive below shows the masses in g, of 90 starfruits.
a) Construct a box plot based on the ogive.
b) Hence, state the distribution shape of the data.
Solution:
a)
\(\begin{aligned} &\text{Minimum value} = 80\\\\ &\text{Maximum value} = 150\\\\ &Q_1=116\\\\ &Q_2=123\\\\ &Q_3=128 \end{aligned}\)
b) The distribution of the data is skewed to the left because the left side of the box plot is longer than the right side of the box plot.
Chapter : Measures of Dispersion of Grouped Data
Topic : Measures of Dispersion
Form 5
Mathematics
View all notes for Mathematics Form 5
Related notes
Dispersion
Direct Variation
Inverse Variation
Joint Variation
Matrices
Basic Operation on Matrices
Risk and Insurance Coverage
Taxation
Congruency
Enlargement
Report this note
Online Tuition
Live class daily with celebrity tutors
Learn more
Register for a free Pandai account now
Edit content
×
Loading...
Quiz
Videos
Notes
Account