**Which of the following is not a measure of dispersion?**

A)Variance

B) Range

C) Arithmetic Mean

D) Standard Deviation

The Correct Answer for the given question is Option **C) Arithmetic Mean**

**Arithmetic Mean is not a measure of dispersion.** Arithmetic mean is not a measure of dispersion because it does not take into account the variability in the data. Scores on a test can be skewed by one or more students who perform extraordinarily well, while scores for other students may be quite low. A better measure of variation would be the standard deviation, which takes into account both high and low scores. Dispersion is best measured by the standard deviation.

Arithmetic mean (or average) is a number that summarizes the central tendency of a set of data. It is calculated by adding up all the values in a dataset and dividing that total by the number of items in the dataset. The arithmetic mean can be used to compare groups of data, to find patterns, or to make comparisons between two datasets.

**Dispersion**

Dispersion plays an very important role for describing the character of variability in data. Only the representative value can be discovered by Average but Dispersion actually finds out how individual value fall apart on an average from the representative value. Some of the major objectives of Dispersion are determining the reliability of central tendency; comparing the consistency of two or more series; determining the cause of variability and controlling it; controlling quality and analyzing the time series.

In statistics, the measure of central tendency gives a single value that represents the whole observation; however, this unique value cannot fully describe the phenomenon. We use the measure of dispersion to analyze the variability of the items. Dispersion in statistics has two meanings: it measures the variation among the items, as well as the variation around the average. Dispersion will be high if there is a large difference between the value and average. If not, it will be low. Researchers use this technique to determine the reliability of averages because it indicates the variation between items, according to Dr. Bowley. Researchers can also compare two or more series using the dispersion. Dispersion is also the facilitating technique for many other statistical techniques, including correlation, regression, structural equation modeling, etc. In statistics, dispersion has two types of measures. The absolute measure determines the dispersion within a statistical unit. As a relative measure, the ratio unit can be used to measure dispersion. In statistics, there are many techniques that are applied to measure dispersion.

**Measures of Dispersion**

A measure of dispersion is a non-negative number that indicates the spread of data about a central value. Dispersion measures indicate how stifled or stretched a given dataset is. Typically, there are five measures of dispersion. This includes range, variance, standard deviation, mean deviation, and quartile deviation. Dispersion measures have the most important use of helping us better understand the distribution of data. The measure of dispersion increases in value as data becomes more diverse. The article discusses dispersion measures, their types, examples, and other important aspects of measuring them.

Some of the various Measures of Dispersion are as follows:

**a) Range**

The difference between Largest and Smallest items in the distribution is called Range. Range is very simple to understand and calculate .Easy calculation is the major advantage of this measure of dispersion. However, it has many disadvantages as well. This method is highly sensitive to outliers and does not consider all observations in a data set. Rather than providing a range of values, it is more informative to provide the minimum and maximum values. The coefficient of Range can be calculated using the formula below:

**Coefficient of Range = (L-S) / (L+S)**

**Merits of Range**

- The simplest way to measure dispersion
- Can be calculated easily
- It is simple to understand
- It does not matter what the origin is

**Demerits of Range**

- Two extreme observations led to this conclusion. As a result, get affected by fluctuations
- A range is not a reliable measure of dispersion
- Dependent on change of scale

**b) Quartile Deviation or Semi-Inter Quartile Range**

Depending upon the lower and upper quartiles , the measure of dispersion is calculated which is called **Quartile Deviation.** **Interquartile Range** is the difference between upper and lower quartile. And Half of the Interquartile Range is called **Semi- Quartile Range. **Quartile Deviation is only the absolute measure of dispersion which indicates that if there is a necessity for comparative study of variability of two distribution then **coefficient of quartile deviation** is needed.

The difference between the 25th and 75th percentiles (also called the first and third quartiles) is the interquartile range. Interquartile ranges depict 50% of observations between the 25th and 75th percentiles. In large interquartile ranges, the middle 50% of observations are generally spread far apart from each other. As a measure of variability, the interquartile range is useful in situations where extreme values are not being recorded precisely (as in the case of intervals of open-ended classes in a frequency distribution). Another advantage is that it is not affected by extreme values. Interquartile range is not amenable to mathematical manipulation, and so it is not a useful measure of dispersion.

**Quartile Deviation = (Q3-Q2)/2**

**Coefficient of Quartile Deviation = (Q3-Q1)/(Q3+Q1)**

**Merits of Quartile Deviation**

- Quartile deviation overcomes all the drawbacks of Range
- Half of the data is used
- Unaffected by changes in origin
- For open-end classification, the best way to measure dispersion

**Demerits of Quartile Deviation**

- 50% of the data is ignored
- Dependent on the change of scale
- An unreliable measure of dispersion

**c) Mean Deviation or Average-Deviation**

The above discussed two terms (Range and Quartile Deviation) are not considered as better measures of dispersion as all of the items are not included in both cases and they do not show variations of the items from an average that ignores the formation of the distribution. But Mean Deviation shows the variation of items from average.

**Merits of Mean Deviation**

- All observations considered
- It provides a minimum value when deviations from the median are taken into account
- Unaffected by changes in origin

**Demerits of Mean Deviation**

- It’s not easy to understand
- Calculating it is difficult and time-consuming
- Depending on the change in scale
- The ignorance of the negative sign creates artificiality and renders further mathematical analysis useless

**d) Standard Deviation**

Standard Deviation is considered as the best measure of dispersion and is free from the defects of other measures of dispersion.A Standard Deviation represents how much a set of values varies. Random variables have a standard deviation equal to the square root of their variance. Standard deviation indicates how near the values are to the mean, while a high standard deviation indicates how dispersed they are from each other.

**Merits of Standard Deviation**

- This method overcomes the drawback of ignoring signs in deviations of the mean
- Suitable for further mathematical analysis
- Observations least affected by fluctuation
- All observations must be constant in order for the standard deviation to be zero
- Unaffected by changes in origin

**Demerits of Standard Deviation**

- Calculation is difficult
- Unlikely to be understood by a layman
- Depending on the change in scale

**e) Coefficient of Variation**

Coefficient of Standard Deviation is the relative measure of dispersion that is based on the standard deviation. Similarly, 100 times the Coefficient of Standard Deviation is called coefficient of variation which is denoted by CV.

**Coefficient of Variation = (Standard Deviation/ Mean) * 100**