# Basic Statistics and Data Analysis

## Median Measure of Central Tendency

Median 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. Median is also considered as a measure of central tendency which divides the data set in two half, where the first half contains 50% observations below the median value and 50% above the median value. If in a data set there are odd number of observations (data points), the median value is the single most middle value after sorting the data set.

Example: Consider the following data set 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3.
To find the median of the given data set, first sort it (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 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.

Example: Consider the following data set, 5, 9, 8, 4, 3, 1, 0, 8, 5, 3, 5, 6, 3, 2.
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 median will be 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 median is preferred measure of central tendency when the data set is skewed or not symmetrical.

For large data set it is relatively very difficult to locate median value in sorted data. It will be helpful to use median value using formula. The formula for 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.

The median formula for even number of observation 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 median measure of central tendency, cannot be found for categorical data.

# Measure of Central Tendency

What is a measure of central tendency and what are the common measures of central tendency? Also when is the median preferred over the mean?

A is the single numerical value considered most typical of the values of a quantitative variable. The most common measures of central tendency are the mode (i.e., the most frequently occurring number), the median (i.e., the middle point or fiftieth percentile), and the mean (i.e., the arithmetic average).

Median is preferred over the mean when the numbers are highly skewed (i.e., non-normally distributed).

# Basics Statistics

The two general rules are

1. If the mean is less than the median, the data are skewed to the left, and
2. If the mean is greater than the median, the data are skewed to the right.

Therefore, if the mean is much greater than the median the data are probably skewed to the right.

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