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Measures of Dispersion - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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  • Dispersion (Range, Mean Deviation) is considered one of the most asked concept.

  • 20 Questions around this concept.

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QuestionMEDIUM

The mean of 5 observations is 5 and their variance is 124.  If three of the observations are 1, 2 and 6 ; then the mean deviation from the mean of the data is :

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If the mean deviation of the numbers 1, 1+d, ..., 1+100 d from their mean is 255, then a value of d is :

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Choose the range and co-efficient of the range of the following data: 63, 89, 98, 125, 79, 108, 117, 68.

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Mean and Standard deviation from the following observations of marks of 5 students of a tutorial group (marks out of 25), 8, 12, 13, 15, 22 are?

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Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable \mathrm{X} denote the number of rotten apples. If \mathrm{\mu \: \: and\: \: \sigma ^{2}} represent mean and variance of \mathrm{X}, respectively, then \mathrm{10(\mu ^{2}+\sigma ^{2})} is equal to

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Let \mathrm{S} be the set of all values of \mathrm{a}_{1} for which the mean deviation about the mean of 100 consecutive positive integers a_{1}, a_{2}, a_{3}, \ldots a_{100} is 25, then S is

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Let the mean of 6 observations 1, 2, 4, 5 x\: and\; y 5, and their variance be 10. Then their mean deviation about the mean is equal to:

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Let the median and the mean deviation about the median of 7 observations 170,125,230,190,210, a, b be 170 and \frac{205}{7} respectively. Then the mean deviation about the mean of these 7 observations is :

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Concepts Covered - 1

Dispersion (Range, Mean Deviation)

Measures of the Spread of the Data

An important characteristic of any set of data is the variation in the data. The degree to which the numerical data tends to vary about an average value is called the dispersion or scatteredness of the data.

The following are the measures of dispersion:

  1. Range

  2. Mean Deviation

  3. Standard deviation and Variance

 

Range

Range is the difference between the highest and the lowest value in a set of observations.

The range of data gives us a rough idea of variability or scatter but does not tell about the dispersion of the data from a measure of central tendency. 

 

Mean Deviation  

Mean deviation for ungrouped data 

 Let n observations are x1 , x2 , x3 , ...., xn .

 If x is a number, then its deviation from any given value a is |x - a|

 To find the mean deviation about mean or median or any other value M of ungrouped data, following steps are involved:

  1. Calculate the measure of central tendency about which we need to find the mean deviation. Let it be 'a' 

  2. Find the deviation of each x_i from a , i.e., |x_1 -a|, \; |x _2 - a|,\; |x_3 -a|,\;\ldots, |x_n - a|

  3. Find the mean of these deviations. This mean is the mean deviation about 'a', i.e.,

          \text{Mean deviation about 'a', }\mathrm{\;\;\;M.D.}(a)=\frac{1}{n}{\sum_{i=1}^{n}\left|x_{i}-a\right|}   

          \text{Mean deviation about mean, }\mathrm{\;\;\;M.D.}(\bar x)=\frac{1}{n}{\sum_{i=1}^{n}\left|x_{i}-\bar x\right|}

          \text{Mean deviation about median, }\mathrm{\;\;\;M.D.}(Median)=\frac{1}{n}{\sum_{i=1}^{n}\left|x_{i}-Median\right|}

 Mean deviation for ungrouped frequency distribution

      Let the given data consist of n distinct values x1 , x2 , ..., xn occurring with frequencies f1 , f2 , ..., fn respectively. 

      \begin{array}{lll}{x: x_{1}} & {x_{2}} & {x_{3} \dots x_{n}} \\ {f: f_{1}} & {f_{2}} & {f_{3} \dots f_{n}}\end{array}

  1. Mean Deviation About Mean 

           First find the mean, i.e.

           \bar{x}=\frac{\sum_{i=1}^{n} x_{i} f_{i}}{\sum_{i=1}^{n} f_{i}}=\frac{1}{\mathrm{N}} \sum_{i=1}^{n} x_{i} f_{i}

           N is the sum of all frequencies

          Then, find the deviations of observations x_i from the mean \bar x and take their absolute values, i.e., \left|x_{i}-\bar{x}\right| \text { for all } i=1,2, \ldots, n

          After this, find the mean of the absolute values of the deviations

          \mathrm{M.D.}(\bar{x})=\frac{\sum_{i=1}^{n} f_{i}\left|x_{i}-\bar{x}\right|}{\sum_{i=1}^{n} f_{i}}=\frac{1}{N} \sum_{i=1}^{n} f_{i}\left|x_{i}-\bar{x}\right|

 

  1. Mean Deviation About any value 'a'

          \mathrm{M.D.}(\mathrm{a})=\frac{1}{\mathrm{N}} \sum_{i=1}^{n} f_{i}\left|x_{i}-\mathrm{a}\right|

 

 Mean deviation for grouped frequency distribution

The formula for mean deviation are the same as in the case of ungrouped frequency distribution. Here, x_i is the mid point of each class. 

 

Note

The mean deviation about the median is the lowest as compared to mean deviation about any other value.

Study it with Videos

Dispersion (Range, Mean Deviation)

Study other Related Concepts

Measures of Dispersion Current Topic

Introduction
Representation of Data
Central Tendency
Measures of Dispersion
Dispersion (Variance and Standard Deviation)
Coefficients of Dispersion
Some Important Point Regarding Statistics
Important Terminologies and Definitions of Probability
Algebra of Events
Basic Probability Practise Session

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