Content Curator
Measures of Dispersion, in statistics, means variability. Thus, Dispersion means the extent to which a central value can be stretched out or how much data is likely to vary around an average value.
- This concept of dispersion in statistics helps in the understanding of the distribution of data.
- Dispersion can also be expressed as the distribution of data.
- The measure of dispersion can never be negative.
- A data set with a large value can be seen to be widely scattered or tightly clustered.
- An example of widely scattered data is: 0, 30, 60, 90, 120, …
- While, for tightly clustered data of small value is: 1, 2, 2, 3, 3, 4, 4,..
Variance, standard deviation, and interquartile are some of the types of dispersion. Dispersion is also called variability, scatter, or spread.
Read Also: Conditional Probability
| Table of Content |
Key Terms: Statistics, Dispersion, Variance, Standard Deviation, Central Tendency, Measures of Dispersion, Distribution, Mean, Median, Mode
Dispersion Meaning
[Click Here for Sample Questions]
Dispersion, in statistics, is a concept that is useful in the study of the distribution of data. Dispersion represents all the possible outcomes for a variable whose distribution is being calculated. It tells us how spread out the data actually is.
- Dispersion can be estimated using several measures, such as mean, standard deviation, variance, and more.
- Statistical dispersion can be defined as the extent to which numerical data tends to vary about an average value.
- Simply, dispersion is known to help understand the distribution of data.
- For example, a data set involving 1,2,3,4…. is a tightly clustered set, and its dispersion isn’t too spread-out.
- While a data set involving 0,10,20,30,40 …. is wider when compared to the former data set and its dispersion is more spread out.
Also read: Frequency Polygon
Solved ExampleExample: Determine the Variance and Standard Deviation of the Following Numbers: 1, 3, 5, 5, 6, 7, 9, 10. Ans: The mean = (1+ 3+ 5+ 5+ 6+ 7+ 9+ 10)/8 = 46/ 8 = 5.75 Step 1: The first step is to Subtract the mean value from the individual value. Thus, (1 – 5.75), (3 – 5.75), (5 – 5.75), (5 – 5.75), (6 – 5.75), (7 – 5.75), (9 – 5.75), (10 – 5.75) = -4.75, -2.75, -0.75, -0.75, 0.25, 1.25, 3.25, 4.25 Step 2: Following that, we have to square the above values, 22.563, 7.563, 0.563, 0.563, 0.063, 1.563, 10.563, 18.063 Step 3: 22.563 + 7.563 + 0.563 + 0.563 + 0.063 + 1.563 + 10.563 + 18.063 = 61.504 Step 4: n = 8, thus the variance (σ2) = 61.504/ 8 = 7.69 Hence, Standard deviation (σ) = 2.77 |
Measures of Dispersion
[Click Here for Previous Year Questions]
Measures of Dispersion definition simply refers to different statistical methods that can be applied to calculate dispersion. Thus, the measures of dispersion definition explain:
| “The disparity of data from one another. The measure of dispersion also denotes the scattering of data.” |
Measures of Dispersion
- It tells us whether the data is homogenous i.e., all the entries in the data set are similar or the data is more diverse.
- This means the entries in the data set are different and far apart, showing more variability.
- The measure of dispersion is always a non-negative and real number.
Read More:
Types of Measures of Dispersion
[Click Here for Sample Questions]
Measures of Dispersion, in statistics, can be divided into:
Absolute Measures of Dispersion
The Absolute Measures of Dispersion include:
- Range: in a given data set whatever the difference between its smallest and largest number is, that is called range. It provides a rough idea of how spread out the data is going to be.
- Variance: it’s defined as the deduction of the mean from each number in the data set then squaring the resultant and adding all those squares together to divide them with the total number of entries in that data set is called variance.
- Quartile and Quartile Deviation: Quartiles are to divide a data set into four equal parts. The Quartile Deviation is half of the distance between the first and the third Quartile where the middle number between the smallest number of data sets and the median of that set is called first Quartile. While, the middle number between the median of the data set and the largest number of the data set is called the third Quartile.
- Standard Deviation: It can be defined as the square root of variance. It always involves positive numbers and can be calculated for all three types of series: individual data series, discrete data series, and continuous data series.
- Mean and Mean Deviation: it is the sum of absolute values of deviations divided by the number of observations. It helps understand the dispersion of data from given measures of central tendency. It calculates the mean, median, and mode of the data separately.
Note: The deviation of an observation ‘p’ from a fixed value ‘k’ is their difference i.e., p – k. But the measure of central tendency lies between maximum and minimum values of the set of observations. So, it includes both positive and negative values which might cancel each other while calculating the mean as their sum becomes zero. That is why the mean deviation isn’t a useful method to calculate measures of Dispersion.
Measures of Dispersion Types
Relative measures of Dispersion
It compares two dispersions in contrast with each other. This comparison does not involve units of the values being compared. Thus, the methods for determining relative measures of Dispersion are:
- Coefficient of Range
- Coefficient of Variation
- Coefficient of Standard Deviation
- Coefficient of Quartile Deviation
- Coefficient of Mean deviation
Coefficient of Variation Detailed Explanation
Coefficient of Dispersion
When two different dispersions with different averages are compared, then the coefficient of dispersion can be calculated by the following methods:
| Co-efficient in terms of | Coefficient of dispersion (formula) |
|---|---|
| Range | C.D. = (Xmax – Xmin) ⁄ (Xmax + Xmin) |
| Quartile Deviation | C.D. = \(\frac{Q_{3}-Q_{1}}{Q_{3}+Q_{1}} \) |
| Standard Deviation (S.D.) | C.D. = S.D. ⁄ Mean |
| Mean Deviation | C.D. = Mean deviation/Average |
Where, C.D is the Coefficient of Dispersion.
Read More: Difference Between Mean, Median, and Mode
Measures of Dispersion Formula
[Click Here for Previous Year Questions]
The formula of Measures of dispersion includes:
Range Formula
The formula of Range is:
| Range = L – S |
- Largest number of the data set (say L)
- Smallest number of the data set (say S)
Variance Formula
The formula of Variance is:
| Variance, (\(\sigma\)2) = ∑(X−μ)2/N |
Where,
- σ2 = Variance
- µ = mean of the data set
- N = number of entries in that data set
- X = individual values given in the data set
So, variance is calculated using the following steps:
Step 1: (xr - µ) i.e. take individual values from the data set and subtract the mean from it.
Step 2: (xr - µ)2 i.e., square all the numbers obtained from the previous step. This is done to remove negative values.
Step 3: Σ (xr - µ)2 i.e., add up all the numbers obtained in the second step.
Step 4: [Σ (xr - µ)2 / n] i.e., divide the result from Step 3 with the number of entries that are in the data set.
Standard Deviation formula
To calculate the standard deviation, first, calculate the variance and then put a square root on the result. Thus, the formula of Standard Deviation is:
| Standard Deviation = √σ2 |
Mean Deviation formula
To calculate Mean Deviation, we need to know that:
Now, the formula of Mean deviation is:
| Mean Deviation = [Σ (xr – a)]/N |
(xr – a) is the deviation of values from the data set “x” from a fixed point “a”. It’s calculated by individually subtracting the value of a fixed point from each of the values given in the data set. Here, N = number of entries in a given data set.
Mean Deviation is of three types:
Individual Data Series: It is when the all the data is given on an individual basis. Thus,
| Item As Given | 22 | 24 | 26 | 28 | 30 | 32 |
|---|
Discrete Data Series: It is when individual data has been accompanied along with its Frequency. Thus,
| Items | 20 | 40 | 60 | 80 |
|---|---|---|---|---|
| Frequency | 9 | 6 | 1 | 4 |
Continuous Data Series: It is when the data is not on an individual basis but the range of data is given along with its frequency.
| Items | 0-5 | 5-10 | 10-20 | 20-30 |
|---|---|---|---|---|
| Frequency | 5 | 5 | 1 | 8 |
Quartile formula
To calculate Quartile deviation i.e., Q, the following formula is used:
| Q = ½ x (Q3 -Q1) |
Where,
- Q3 = [3(n+1)]/4 th term
- And Q1 = (n + 1)/4 th term
Thus, it can be summed as:
| Absolute Measures of Dispersion | Formula |
|---|---|
| Range | H - S (H = Highest Value, S = Smallest Value in a data set) |
| Variance | Population Variance, \(\sigma^2 = \sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n} \), Sample Variance, s2 = \(\sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n-1} \) |
| Standard Deviation | Population Standard Deviation, SD = \(\sqrt{Variance} = \sigma \), Sample Standard Deviation, SD = s |
| Mean Deviation | \(\sum_{1}^{n}\frac{|X - \overline{X}|}{n} \) (X = Central Tendency) |
| Quartile Deviation | \(\frac{Q_{3}-Q_{1}}{2} \) (Q3 and Q1 are the third and first quartiles respectively) |
Standard Deviation Calculation
[Click Here for Sample Questions]
There are three methods of calculating the standard deviation:
Direct method
Here, the arithmetic mean is to be evaluated using the formula.
- Following that, the deviations of all the observations from this mean value are to be calculated.
- Then these deviations are to be squared with their addition divided by the number of observations.
- Thus, the square root of the calculation can help result in the derivation of the standard deviation.
Short-cut method
In this method, any random value for the calculation of deviation is assumed such that it lies around the middle of the range of values. Upon considering an extreme value, the deviations would be large with longer calculations.
Step-deviation method
The step deviation method is also called an extension or the simplification of the shortcut method.
- It can be done by picking a common factor between deviations such that when the value is divided by the factor, all the deviation values are further reduced to a simple number.
- This form of reduction makes the calculation easier.
- This is why the Step-deviation method is preferred more than the other two methods.
Also Read:
| Chapter-Related Topics | ||
|---|---|---|
| Chi-Square Formula | T-Test Formula | Standard Error Formula |
| Skewness Formula | Median | Coefficient of Variation Formula |
Measures of Dispersion and Central Tendency
[Click Here for Previous Year Questions]
Both measures of dispersion and measures of central tendency can be used in order to represent data. Thus, the difference between the measures of dispersion and central tendency are:
| Measures of Dispersion | Central Tendency |
|---|---|
| In order to quantify the variability of data, measures of dispersion are used. | Measures of central tendency are used to quantify the data's average behaviour. |
| Measures of dispersion contain variance, mean deviation, standard deviation, quartile deviation, and more. | Measures of central tendency contain mean, median, and mode. |
Things to Remember
- Dispersion, as a statistical concept, is the measure of how spread out the given set of data is.
- To calculate dispersion, the method of calculating variance and the standard deviation is most preferred.
- Dispersion can be defined as the extent to which a central value enables to be stretched out or how much data tends to vary around an average value.
- Measures of Dispersion is of three types: Absolute Measures of Dispersion, Relative Measures of Dispersion, and Coefficient of Dispersion.
- The formula of Variance is (\(\sigma\)2) = ∑(X−μ)2/N.
Also Read: Interquartile Range
Previous Year Questions
- The mean and variance of 7 observations are 8 and…
- The mean and variance of 20 observations are found to… (JEE Main 2020)
- The variance of the numbers 2, 3, 11 and x is…
- Two cards are drawn at random from a pack of 52 cards ….(KCET 2016)
- Probability of four sons to a couple is…..(NEET 2001)
- If the mean and the variance of a binomial variate… (JEE Main 2015)
- The variance of first n even natural numbers is… (JEE Main 2009)
- A measure of dispersion is indicative of the reliability…
- The mean and standard deviation of marks obtained…
- The variance of the first 50 even natural numbers is… (JEE Main 2014)
- The variance of the first 20 natural numbers is…
Sample Questions
Ques. Find the range of the given data set: 4, 8, 19, 10, 2, 13, 17. (2 marks)
Ans. The formula for Range is: R = largest number – smallest number.
So, R = 17 – 2
R = 15.
Ques. Calculate standard deviation for the data set 5, 9 12. (2 marks)
Ans. Standard deviation = √σ2
Variance, s2 = S (xr - m)2 / n
here, mean (µ) = [5 + 9 + 12]/3 = 26/3 = 8.6 and n =3
Σ (xr - µ)2 = (5- 8.6)2 + (9 – 8.6)2 + (12 – 8.6)2 = 12.96 + 0.16 + 11.56 = 24.68
σ 2 = 24.68/3 = 8.22
Standard Deviation(σ) = √8.22 = 2.86.
Ques. Find the Quartile Deviation of the data set 17, 18, 24. (3 marks)
Ans. Quartile Deviation, Q = ½ x (Q3 – Q1)
Where, number of enters in the data set, n = 3
Q3 = [3(n+1)]/4th term
Q3 = [3(3+1)]/4 = 12/4 = 3rd term which is 24.
And Q1 = (n + 1)/4th term
Q1 = (3+1)/4 = 4/4 = 1st term which is 17
Q = ½ X (24 -17) = ½ X 7 = 3.5
Ques. Why is Dispersion important in Statistics? (1 mark)
Ans. Dispersion is important in statistics to determine if the data is spread out or is tightly clustered. Besides, it also helps to find the consistency of performance in statistics.
Ques. Determine the range and coefficient of range of the given data values: 45, 55, 63, 76, 67, 84, 75, 48, 62, 65. (3 marks)
Ans. Assume Xi values to be:
45, 55, 63, 76, 67, 84, 75, 48, 62, 65
Thus, here,
Maxium value (Xmax) = 84
And, Minimum or Least value (Xmin) = 45
Range = Maximum value = Minimum value
= 84 – 45
= 39
Hence,
The Coefficient of range = (Xmax – Xmin)/(Xmax + Xmin)
= (84 – 45)/(84 + 45)
= 39/129
= 0.302 (approximate value)
Ques. Determine the sample variance of the given data set {2, 6, 12, 15}. (3 marks)
Ans. Variance is a measure of dispersion. Thus, it can be denoted by:
= \(\sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n-1} \)
Hence, n = 4
\(\bar{X}\) = (2 + 6 + 12 + 15) / 4 = 8.75
Variance = \(\frac{(2 - 8.75)^{2} + (6 - 8.75)^{2} + (12 - 8.75)^{2} + (15 - 8.75)^{2}}{3} \)
= 34.25
Ques. Determine the range and coefficient of range of the given data set {8, 12, 5, 6, 8, 2,15}. (2 marks)
Ans. Range can be defined as a measure of dispersion given by the Highest value (H) - Smallest value(S)
- H = 15,
- S = 2
Thus, Range = 15 - 2 = 13
Coefficient of Range = (H - S) / (H + S)
= 13 / 17
= 0.76
Ques. What are the objectives of Measures of Dispersion? (3 marks)
Ans. The objectives of measures of dispersion include:
- It helps to determine the average distance of data points from the average of the data set.
- It also helps to identify the variability in data.
- It also helps to compare two or more data sets based on variability.
Ques. What are the several advantages and disadvantages of Measures of Dispersion? (4 marks)
Ans. The advantages and disadvantages of the measures of dispersion include:
Advantages of Measures of Dispersion:
- It helps to identify the reliability of the average value of the given data set.
- It also helps to quantify the variability or dispersion of the data points present in a data set.
Disadvantages of Measures of Dispersion:
- The computation process of particular measures of dispersion can potentially be difficult and lengthy.
- It does not provide an idea of symmetricity.
Ques. What is Discrete Data Series? List its Example. (2 marks)
Ans. Discrete Data series is when individual data has been followed by its Frequency. An example of Discrete data series is.
| Items | 20 | 40 | 60 |
|---|---|---|---|
| Frequency | 9 | 6 | 1 |
Also Read:
Comments