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Dispersion (Range, Mean Deviation) is considered one of the most asked concept.
20 Questions around this concept.
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 :
If the mean deviation of the numbers 1, 1+d, ..., 1+100 d from their mean is 255, then a value of d is :
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?
Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable denote the number of rotten apples. If represent mean and variance of , respectively, then is equal to
Let be the set of all values of for which the mean deviation about the mean of 100 consecutive positive integers is 25, then is
Let the mean of 6 observations 1, 2, 4, 5 5, and their variance be 10. Then their mean deviation about the mean is equal to:
Let the median and the mean deviation about the median of 7 observations 170,125,230,190,210, a, b be 170 and respectively. Then the mean deviation about the mean of these 7 observations is :
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:
Range
Mean Deviation
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 x_{1} , x_{2} , x_{3} , ...., x_{n} .
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:
Calculate the measure of central tendency about which we need to find the mean deviation. Let it be
Find the deviation of each from , i.e.,
Find the mean of these deviations. This mean is the mean deviation about , i.e.,
Mean deviation for ungrouped frequency distribution
Let the given data consist of n distinct values x_{1} , x_{2} , ..., x_{n} occurring with frequencies f_{1} , f_{2} , ..., f_{n} respectively.
Mean Deviation About Mean
First find the mean, i.e.
N is the sum of all frequencies
Then, find the deviations of observations from the mean and take their absolute values, i.e.,
After this, find the mean of the absolute values of the deviations
Mean Deviation About any value 'a'
Mean deviation for grouped frequency distribution
The formula for mean deviation are the same as in the case of ungrouped frequency distribution. Here, 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.
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