# Relationship between the mean mode and median

The statistical mean, median, mode and range for data informs users of variation, changes over time, or outliers and acceptable norms. An empirical relationship exists between mean, median and mode. For a moderately skewed distribution it is: image. If a frequency distribution. The relationship between the mean, median, and mode is shown with these 3 histograms. Symmetric histogram Skewed-to-the-left histogram.

Further, since the total number of observations will also increase, the median would also be affected but to a lesser extent than mean.

Finally, there would be no change in the position of mode.

## Measures of Central Tendency

Empirical Relationship between Mean, Median and Mode Empirical Relation between Mean, Median and Mode Empirically, it has been observed that for a moderately skewed distribution, the difference between mean and mode is approximately three times the difference between mean and median, i. The mean and median of a moderately skewed distribution are Find mode of the distribution. For a moderately skewed distribution, the median price of men's shoes is Rs and modal price is Rs Calculate mean price of shoes.

Choice of a Suitable Average The choice of a suitable average, for a given set of data, depends upon a number of considerations which can be classified into the following broad categories: Considerations based on the suitability of the data for an average. Considerations based on the purpose of investigation.

Considerations based on various merits of an average. The nature of the given data may itself indicate the type of average that could be selected. For example, the calculation of mean or median is not possible if the characteristic is neither measurable nor can be arranged in certain order of its intensity.

However, it is possible to calculate mode in such cases. Suppose that the distribution of votes polled by five candidates of a particular constituency are given as below: Since the above characteristic, i. However, the mode of the distribution is D and hence, it can be taken as the representative of the above distribution. If the characteristic is not measurable but various items of the distribution can be arranged in order of intensity of the characteristics, it is possible to locate median in addition to mode.

For example, students of a class can be classified into four categories as poor, intelligent, very intelligent and most intelligent.

Here the characteristic, intelligence, is not measurable. However, the data can be arranged in ascending or descending order of intelligence. It is not possible to calculate mean in this case.

If the characteristic is measurable but class intervals are open at one or both ends of the distribution, it is possible to calculate median and mode but not a satisfactory value of mean. However, an approximate value of mean can also be computed by making certain an assumption about the width of class es having open ends.

If the distribution is skewed, the median may represent the data more appropriately than mean and mode. If various class intervals are of unequal width, mean and median can be satisfactorily calculated. However, an approximate value of mode can be calculated by making class intervals of equal width under the assumption that observations in a class are uniformly distributed.

The accuracy of the computed mode will depend upon the validity of this assumption. The choice of an appropriate measure of central tendency also depends upon the purpose of investigation. If the collected data are the figures of income of the people of a particular region and our purpose is to estimate the average income of the people of that region, computation of mean will be most appropriate. On the other hand, if it is desired to study the pattern of income distribution, the computation of median, quartiles or percentiles, etc.

Similarly, by calculating quartiles or percentiles, it is possible to know the percentage of people having at least a given level of income or the percentage of people having income between any two limits, etc.

If the purpose of investigation is to determine the most common or modal size of the distribution, mode is to be computed, e. The computation of mean and median will provide no useful interpretation of the above situations. The presence or absence of various characteristics of an average may also affect its selection in a given situation. If the requirement is that an average should be rigidly defined, mean or median can be chosen in preference to mode because mode is not rigidly defined in all the situations.

An average should be easy to understand and easy to interpret.

**03 Relationship of Mean, Median, Mode**

When you have an even number of points, you must determine the two central positions of the data set. See side box for instructions. Looking at our dataset, we see that the 4th and 5th numbers are 14 and From there, we return to our trusty friend the mean to determine the median. Mode[ edit ] The mode is the most common or "most frequent" value in a data set.

### Statistics/Summary/Averages/mean - Wikibooks, open books for an open world

This is the most common value of the data set. Data sets having one mode are said to be unimodal, with two are said to be bimodal and with more than two are said to be multimodal.

The mode for this data set is 4. This is because both 2 and 3 are modes. If all points in a data set occur with equal frequency, it is equally accurate to describe the data set as having many modes or no mode.

Midrange[ edit ] The midrange is the arithmetic mean strictly between the minimum and the maximum value in a data set. Relationship of the Mean, Median, and Mode[ edit ] The relationship of the mean, median, and mode to each other can provide some information about the relative shape of the data distribution. If the mean, median, and mode are approximately equal to each other, the distribution can be assumed to be approximately symmetrical.