Basic Statistics

The post is about the Formula of Median, its definition, and examples for the computation of the median for an even or odd number of observations in a data set.

Introduction of Median

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 a measure of central tendency that divides the data set into two halves, where the first half contains 50% observations below the median value and 50% above the median value. If there are an odd number of observations (data points) in a data set, the median value is the single-most middle value after sorting the data set.

After understanding the median definition, let’s consider a few examples of how 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. After sorting, the middle-most value of the above data 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.

Formula of Median 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 compute the median value using the formula. The median 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.

Formula of Median for Even Number of Observations

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

The computation/ calculation of the median is a crucial step in exploratory data analysis (EDA). It helps identify potential outliers, assess skewness in the data distribution, and choose appropriate statistical methods for further analysis.

Applications of Median in Different Scenarios

1. Resisting Outliers: The median’s primary strength lies in its resistance to outliers. Unlike the mean (which can be swayed by extreme values), the median remains unaffected and stable by a few very high or very low data points (extreme observations).

2. Analyzing Skewed Distributions: When dealing with data that is not symmetrical (has skewed distributions), the median provides a more accurate representation of the “center” of the data compared to the mean/average. The median reflects the value that divides the data into halves, whereas the mean gets pulled towards the tail of the skewed distribution.

3. Ease of Interpretation: The median is a simple concept – the middle (centermost) value when the data is arranged in order (either ascending or descending).

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

FAQs about Median

1. What is the median?
2. What is the advantage of the median over other measures of central tendencies?
3. On what kind/type of data median can be computed?
4. What is the benefit of using the median?
5. What is the formula of median when the number of observations is even and when the number of observations is odd?
6. How median is interpreted?
7. In how many groups median classify the data/sample/population?

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R Programming Language

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.

Weight of 10 year child	Frequency
22	1
23	5
25	2
30	3
32	1
33	2
Total	15

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

Weight of 10 year child	Frequency
Male	11
Female	9
Total	25

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

Group	Values	Frequency
0 to 9	3, 4, 7	3
10 to 19	11, 13, 15	3
20 to 29	20, 22, 23, 25, 26	5
30 to 39	33	1
Total	12

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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