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# Difference Between Median and Average (Mean)

Median vs Average (Mean)

Median and mean are measures of central tendency in descriptive statistics. Often Arithmetic mean is considered as the average of a set of observations. Therefore, here mean is considered as the average. However, average is not the arithmetic mean at all the times.

Average

The arithmetic mean is the sum of the data values divided by the number of data values, i.e.

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_{i} = \frac{x_{1}+x_{2}+x_{3}+...+x_{n}}{n}$

If the data is from a sample space it is called a sample mean ($\bar{x}$), which is a descriptive statistic of the sample. Although it is the most commonly used descriptive measure for a sample, it is not a robust statistic. It is very sensitive to the outliers and oscillations.

For example, consider the average income of the citizens of a particular city. Since all the data values are summed and then divided, the income of an extremely wealthy person affects the mean significantly. Therefore, the mean values are not a good representation of the data always.

Also, in the case of an alternating signal, the current passing through an element periodically varies from the positive direction to negative direction and vice versa. If we take the average current passing through the element in a single period, it will give a 0, meaning that no current has passed through the element, which obviously is not true. Therefore, in this case too, arithmetic mean is not a good measure.

The arithmetic mean is a good indicator when the data is evenly distributed. For a normal distribution, the mean is equal to the mode and median. It also has the lowest residuals when considering the root mean squared error; therefore, the best descriptive measure when it is required to represent a dataset by a single number.

Median

The values of the middle data point after arranging all the data values in ascending order is defined as the median of the dataset.

• If the number of observations (data points) is odd, then the median is the observation exactly in the middle of the ordered list.

• If the number of observations (data points) is even, then the median is the mean of the two middle observations in the ordered list.

Median divides the observation into two groups; i.e. a group (50%) of values higher and a group (50%) of values lower than the median. Medians are specifically used in skewed distributions and represent data fairly better than the arithmetic mean.

Median vs Mean (Average)

• Both mean and median are measures of central tendency and summarize the data. Mean is independent of the position of the data points, but the median is calculated using the position.

• Mean is heavily affected by outliers while the median is not affected.

• Therefore, median is a better measure than the mean in the cases of highly skewed distributions.

• In the standard, normal distributions, the means and median are the same.

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