# Measure of Central Tendency

Measures of Central Tendency are used to find the center of a data set. T These include measure of location, measure of position, and measure of center. Mean, median, mode, weighted mean, harmonic mean, geometric mean, percentiles, quartiles, deciles, fractiles, etc., are all examples of measures of central tendencies.

## Median Definition, Formula, and Example

### Median Definition

Median (a measure of central tendency) is the middle-most value in the data set when all of the values (observations) in a data set are arranged either in ascending or descending order of their magnitude. The median is also considered as a measure of central tendency which divides the data set in two halves, where the first half contains 50% observations below the median value and 50% above the median value. If in a data set, there are an odd number of observations (data points), the median value is the single-most middle value after sorting the data set.

After understanding the median definition, let us consider few examples to calculate the median for a data set.

#### Median Example – 1

Question: For the following data set: 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3, calculate the median.

Answer: To find the median of the given data set, first sort the data (either in ascending or descending order), that is
0, 1, 3, 3, 3, 4, 5, 5, 5, 6, 8, 8, 9. The middle-most value of the above data after sorting is 5, which is the median of the given data set.

When the number of observations in a data set is even then the median value is the average of two middle-most values in the sorted data.

#### Median Example – 2

Question: Consider the following data set, 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3, 2. Compute the median.

Answer: To find the median first sort it and then locate the middle-most two values, that is,
0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 8, 8, 9. The middle-most two values are 4 and 5. So the median will be the average of these two values, i.e. 4.5 in this case.

The median is less affected by extreme values in the data set, so the median is the preferred measure of central tendency when the data set is skewed or not symmetrical.

### Median Formula for Odd Number of Observations

For large data sets it is relatively very difficult to locate median values in sorted data. It will be helpful to use the median value using the formula. The formula for an odd number of observations is
\begin{aligned} Median &=\frac{n+1}{2}th\\ Median &=\frac{n+1}{2}\\ &=\frac{13+1}{2}\\ &=\frac{14}{2}=7th \end{aligned}

The 7th value in sorted data is the median of the given data.

### Median Formula for Even Number of Observations

The median formula for an even number of observations is
\begin{aligned} Median&=\frac{1}{2}(\frac{n}{2}th + (\frac{n}{2}+1)th)\\ &=\frac{1}{2}(\frac{14}{2}th + (\frac{14}{2}+1)th)\\ &=\frac{1}{2}(7th + 8th )\\ &=\frac{1}{2}(4 + 5)= 4.5 \end{aligned}

Note that the median measure of central tendency, cannot be found for categorical data.

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## Mode Measure of Central Tendency

The mode is the most frequent observation in the data set i.e. the value (number) that appears the most in the data set. It is possible that there may be more than one mode or it may also be possible that there is no mode in a data set. Usually, it is calculated for categorical data (data belongs to nominal or ordinal scale) but is unnecessary.

It can also be used for ordinal and ratio scales, but there should be some repeated value in the data set or the data set can be classified. If any of the data points don’t have the same values (no repetition in data values), then the mode of that data set will not exit or may not be meaningful. A data set having more than one mode is called multimode or multimodal.

Example 1: Consider the following data set showing the weight of a child at the age of 10 years: 33, 30, 23, 23, 32, 21, 23, 30, 30, 22, 25, 33, 23, 23, 25. We can find the most repeated value by tabulating the given data in the form of a frequency distribution table, whose first column is the weight of the child and the second column is the number of times the weight appears in the data i.e. frequency of each weight in the first column.

From the above frequency distribution table, we can easily find the most repeated occurring observation (data point), which will be the mode of the data set and it is 23, meaning that the majority of the 10-year-old children weigh 23kg. Note that for finding the mode it is not necessary to make a frequency distribution table, but it helps in finding the mode quickly and the frequency table can also be used in further calculations such as percentage and cumulative percentage of each weight group.

Example 2: Consider we have information about a person about his/her gender. Consider the $M$ stands for male and $F$ stands for Female. The sequence of the person’s gender noted is as follows: F, F, M, F, F, M, M, M, M, F, M, F, M, F, M, M, M, F, F, M. The frequency distribution table of gender is

The most repeated gender is male, showing that the most frequent or majority of the people have male gender in this data set.

Mode can be found by simply sorting the data in ascending or descending order and then counting the frequent value without sorting the data especially when data contains a small number of observations, though it may be difficult to remember the number of times which observation occurs. Note that the mode is not affected by the extreme values (outliers or influential observations).

The mode is also a measure of central tendency, but it may not reflect the center of the data very well. For example, the mean of the data set in example 1, is 26.4kg while the mode is 23kg. Therefore, it should be used, if it is expected that data points will repeat or have some classification in them. For such kind of data, one should use it as a measure of central tendency instead of mean or median. For example,

• In the production process, a product can be classified as a defective or non-defective product.
• Student grades can classified as A, B, C, D, etc.
• Gender of respondents
• Blood Group

Example 3: Consider the following data. 3, 4, 7, 11, 15, 20, 23, 22, 26, 33, 25, 13. There is no mode of this data as each value occurs once. By grouping this data in some useful and meaningful form we can get the most repeated value of the data for example, the grouped frequency table is

We cannot find the most Frequent value from this table, but we can say that “20 to 29” is the group in which most of the observations occur. We can say that this group contains the mode which can be found by using the grouped formula.

### Mode from Bar Graph

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## A measure of Central Tendency

The Measure of central tendency is a statistic that summarizes the entire quantitative or qualitative set of data in a single value (a representative value of the data set) tending to concentrate somewhere in the center of the data. The tendency of the observations to cluster in the central part of the data is called the central tendency and the summary values as measures of central tendency, also known as the measure of location or position, are also known as averages.

Note that

• The Measure of central tendency should be somewhere within the range of the data set.
• It should remain unchanged by a rearrangement of the observations in a different order.

### Criteria of Satisfactory Measures of Location or Averages

There are several types of averages available to measure the representative value of a set of data or distribution. So an average should satisfy or possess all or most of the following conditions.

• It should be well defined i.e. rigorously defined. There should be no confusion in its definition. Such as Sum of values divided by their total number is the well-defined definition of Arithmetic Mean.
• It should be based on all the observations made.
• Should be simple to understand and easy to interpret.
• Can be calculated quickly and easily.
• Should be amenable/manageable to mathematical treatment.
• Should be relatively stable in repeating sampling experiments.
• Should not be unduly influenced by abnormally large or small observations (i.e. extreme observations)

The mean, median, and mode are all valid measures of central tendencies, but under different conditions, some measures of central tendencies become more appropriate to use than others. There are several different kinds of calculations for central tendency where the kind of calculation depends on the type of the data i.e. level of measurement on which data is measured.

#### A measure of Central Tendency

The following are the measures of central tendencies for univariate or multivariate data.

• The arithmetic mean: The sum of all measurements divided by the number of observations in the data set
• Median:  The middlemost value for sorted data. The median separates the higher half from the lower half of the data set i.e. partitioning the data set into parts.
• Mode: The most frequent or repeated value in the data set.
• Geometric mean: The nth root of the product of the data values.
• Harmonic mean: The reciprocal of the arithmetic mean of the reciprocals of the data values
• Weighted mean: An arithmetic mean incorporating the weights to elements of certain data.
• Distance-weighted estimator: The measure uses weighting coefficients for $x_i$ that are computed as the inverse mean distance between $x_i$ and the other data points.
• Truncated mean: The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
• Midrange: The arithmetic mean of the maximum and minimum values of a data set.
• Midhinge: The arithmetic mean of the two quartiles.
• Trimean: The weighted arithmetic mean of the median and two quartiles.
• Winsorized mean: An arithmetic mean in which extreme values are replaced by values closer to the median.

Reference:
1) Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
2) http://en.wikipedia.org/wiki/Central_tendency
3) Dodge, Y. (2005) The Concise Encyclopedia of Statistics. Springer,

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