“Measures of Dispersion”.
Types of data.
Measures of Dispersion.
Suitability of a dispersion method.
Types of data
1- Qualitative Data 2- Quantitative Data
1- Qualitative Data
● Uses words and descriptions.
● Can be observed.
● Examples of qualitative data: descriptions of texture, taste, or
2- Quantitative Data
● Expressed with numbers.
● Can be put into categories, measured, or ranked.
● Examples of quantitative data: length, weight, age, cost, salary.
The Quantitative Data has two types:
1- Categorical data 2- Continuous data
- has been placed into groups.
- Example: hair color, opinions.
- numerical data measured on
a continuous range or scale.
- Example: height, weight.
What might be the qualitative and quantitative data that
describe this cup of coffee?
- The qualitative data: it has a strong taste and robust aroma.
- The quantitative data: it is 150 degrees Fahrenheit, and costs 10 SR.
MEASURES OF DISPERSION
In addition to the measures of central tendency such as mean, mode,median we often
need to calculate a second type of measure called a measure of dispersion which
measures the variation in the observations about the middle value– mean or
● The measure of central tendency of any series or data distribution summarises it
into single representative for which are useful in many respect but it fails to account
the general distribution pattern of data.
● Thus any conclusion only based on central tendency may be misleading
● Dispersion can prove very effective in association with central tendency in making
any statistical decision.
WHAT IS MEASURES OF DISPERSION ?
Measures of dispersion: group of analytical tools that describes the spread or
variability of a data set
Suppose that we have the distribution of the yields (kg per plot) of two paddy
varieties from 5 plots each. The distribution may be as follows:
Variety I 45 42 42 41 40 Variety II 54 48 42 33 30
It can be seen that the mean yield for both varieties is 42 kg but cannot say
that the performances of the two varieties are same. There is greater uniformity of
yields in the first variety whereas there is more variability in the yields of the
second variety. The first variety may be preferred since it is more consistent in yield
It is the value of dispersion which says how much reliable a central tendency is?
Usually, a small value of dispersion indicates that measure of central tendency is more
reliable representative of data series and vice‐versa.
There are different measures of dispersion like the range, the quartile deviation, the
mean deviation and the standard deviation
IMPORTANCE OF MEASURES OF DISPERSION
● supplements an average or a measure of central tendency.
● compares one group of data with another.
● Indicates how representative the data is.
● Measure of dispersion is also used to compare uniformity of different data like income,
temperature, rainfall, weight, height… etc.
The range in statistics is the difference between the
maximum and minimum values of a data set. We will
learn more details concerning this very basic descriptive
Uses of Range
With all its limitations Range is commonly used in certain fields. These are:
(i) Quality Control:
In quality control of manufactured products, range is used to study the variation in the
quality of the units manufactured. Even with the most modern mechanical equipment
there may be a small, almost insignificant, difference in the different units of a
commodity manufactured. Thus, if a company is manufacturing bottles of a particular
type, there may be a slight variation in the size or shape of the bottles manufactured.
In such cases a range is usually determined, and all the units which fall within these
limits are accepted while those which fall outside the limits are rejected.
Uses of Range
(ii) Variation in Money Rates, Share values, Exchange Rates and Gold prices, etc:
Variations in money rates, share values, gold prices and exchange rates are commonly
studied through range because the fluctuations in them are not very large. In fact
range as a measure of dispersion should be generally used only when variations in the
value of the variable are not much.
(iii) Weather forecasting:
Range gives an idea of the variation between maximum and minimum levels of
temperature. From day to day the range would not vary much and it is helpful in
studying the vagaries of nature if variations suddenly rise or fall.
Quartiles in statistics are values that divide your data into quarters. they divide your
data into four segments according to where the numbers fall on the number line.
The four quarters that divide a data set into quartiles are:
1. The lowest 25% of numbers.
2. The next lowest 25% of numbers (up to the median).
3. The second highest 25% of numbers (above the median).
4. The highest 25% of numbers.
How can you find the quartile ?
If a data set of scores is arranged in ascending order of magnitude, then:
The lower quartile (Q1) is the median of the lower half of the data set.
Q 2 (The median) is the middle value of the data set
The upper quartile (Q3) is the median of the upper half of the data set.
semi-inter-quartile range or the quartile deviation:
Quartile Deviation (QD) means the semi variation between the upper quartiles (Q3) and lower quartiles (Q1)
in a distribution
the inter quartile rang (IQR) : is the spread of the middle 50% of the data values. So:
Coefficient of Quartile Deviation:A relative measure of dispersion based on the quartile deviation is called
the coefficient of quartile deviation.
Quartile Deviation For Ungrouped Data
First: Arrange the data in ascending order.
Second: Find First Quartile
Third: Find Third Quartile
Finally: Put the values into the Formula of Quartile Deviation
Problem: Following are run scores by batsman in last 20 test
First: Arrange the data in Ascending order
Quartile Deviation For grouped Data
Calculate the quartile deviation and coefficient of quartile deviation from the
data given below:
Quartile Deviation For grouped Data
We have to calculate:
● Class boundaries :
The lower limit for every class is the smallest value in that class.
the upper limit for every class is the greatest value in that class.
The size of the gap between classes is the difference between the
upper class limit of one class and the lower class limit of the next
In this case the gap is
9.8-9.7 = 0.1
The lower boundary of each class is calculated by subtracting half of the gap
9.3 - (0.1 / 2 ) =9.25
the upper boundary of each class is calculated by adding half of the gap value
9.7 + (0.1 /2 ) = 9.75
Quartile Deviation For grouped Data
We have to calculate:
● Cumulative frequency
The total of a frequency and all
frequencies so far in a frequency
It is the 'running total' of frequencies.
Mean Deviation is the arithmetic mean of the differences of the values
from their average. The average used is either the arithmetic mean or
Since the average is a central value, some deviations are positive and
some are negative. If these are added as they are, the sum will not
reveal anything; because the sum of deviations from Arithmetic Mean
is always zero.
Mean Deviation tries to overcome this problem by ignoring the signs of
deviations, i.e., it considers all deviations positive.
Mean Deviation for ungrouped data.
Direct Method Steps:
The A.M. of the values is calculated.
Difference between each value and the A.M. is calculated. All differences are
considered positive. These are denoted as |d|
The A.M. of these differences (called deviations) is the Mean Deviation.
i.e. MD = Σ | d |
Calculate the Mean Deviation of the
following values; 2, 4, 7, 8 and 9.
Mean Deviation for Grouped Data
1. Calculate the mean of the distribution.
2. Calculate the absolute deviations |d| of the class midpoints from the mean.
3. Multiply each |d| value with its corresponding frequency to get f|d| values. Sum
them up to get ∑f∣d∣.
4. Apply the following formula,
M.D. = ∑f∣d∣
Calculate the mean deviation of the following distribution:
Profits of Companies (Rs in lakhs) Number of Companies
The Standard Deviation is a number that measures how far away each number in a
set of data is from their mean ,it shows the variation in data.
Standard Deviation is also known as root-mean square deviation as it is the square
root of means of the squared deviations from the arithmetic mean.
If the Standard Deviation is large it means the numbers are spread out from their
mean. If the Standard Deviation is small it means the numbers are close to their
The symbol for Standard Deviation used for a sample data is s and the symbol used
for a population data is σ.
Formulas of Standard Deviation
In case of individual observations, Standard Deviation can be computed in any of the two ways:
1. Take the deviation of the items from the actual mean
2. Take the deviation of the item from the assumed mean
The formula for an ungroup data is:
´The formula for an ungroup data using assumed mean method:
Step1 take the square of the values
Step 2 take square of each value
Step 3 take sum of both columns
Step 4 Apply them to the formula
S= ⎷(83/3 - (15)2/9)
Where Standard deviation is Applied
Investment firms use Standard deviation for their mutual funds.
In financial terms, standard deviation is used to measure risk involved in
In physical experiments, it is important to have a measurement of
uncertainty. Standard deviation provides a way to check the results.
Very large values of standard deviation can mean the experiment is
Web Analytics will help you get an idea of just how important events that
happen on your website could be using Standard deviation.
grouped data standard Deviation
M = mid-point
μ = Mean
F = frequency
n= number of samples
σ2 = data variance
σ= standard deviation
Find an estimate of the variance
and standard deviation of the
following data for the marks
obtained in a test by 88 students.
Marks Frequency (f)
0 ≤ x < 10 6
10 ≤ x < 20 16
20 ≤ x < 30 24
30 ≤ x < 40 25
40 ≤ x < 50 17
step 1:find the mid-point for each group or range of the frequency table.
(0+10)/2 = 5
step 2: calculate the number of samples of a data set by summing up the
n= 6+16+24+25+17= 88
step 3: find the mean for the grouped data by dividing the addition of
multiplication of each group mid-point and frequency of the data set by the
number of samples.
μ= (5×6 + 15×16 + 25×24 + 35×25 + 45×17)/n
μ= 2510/88 = 28.5227
Step 4: calculate the variance for the frequency table data.
σ2=Σ(F × M2) - (n × μ2)/(n - 1)
σ2= (6×52 + 16×152 + 24×252 + 25×352 + 17×452) - (88 × (28.5227)2)/(88-1)
step 5:estimate standard deviation for the frequency table by taking square
root of the variance.
Solve the case
Find the following measures:
2. Mean deviation
3. Standard deviation
4. Quartile deviation
Use the following formulas to solve the problem
2. Quantile deviation=Q3-Q1/2
Grouped Q1= L+((N/4*CF)/F)i
Suitability of a dispersion method
Range is the simplest method of studying dispersion. Range
is the difference between the smallest value and the largest
value of a series. While computing range, we do not take into
account frequencies of different groups.
Formula: Absolute Range = L – S
Coefficient of Range =
where, L represents largest value in a distribution
S represents smallest value in a distribution
•Shows how much data is clustered around a mean value
•It gives a more accurate idea of how the data is distributed
•Not as affected by extreme values
•It doesn't give you the full range of the data
•It can be hard to calculate
•Only used with data where an independent variable is plotted against the frequency of it
•Assumes a normal distribution pattern
The mean deviation is actually more efficient than the standard
deviation in the realistic situation where some of the
measurements are in error, more efficient for distributions other
than perfect normal, closely related to a number of other useful
analytical techniques, and easier to understand.
Form the above example it is obvious that a measure of central tendency alone is not sufficient to describe a frequency distribution. In addition to it we should have a measure of scatterness of observations. The scatterness or variation of observations from their average are called the dispersion.