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Difference Between Mean and Median
Statistics is a branch of math that deals with the concepts of mean and median. Mean is the average of a certain number of items in a data set, whereas, median is the middlemost value in a given data set. Mean, median, and mode are the measures of central tendency. The central value represents the full set of values present in a data set. It is found by adding up all the data values and dividing them by the total number of values in the data set. For example, the mean of the first five natural numbers is (1+2+3+4+5) / 5 which is 3. The median for the first five natural numbers is also 3, since 3 is the central value in the data set. The measure of the central tendency to be chosen depends upon the type of data set.
Mean Definition
Mean is the ratio of the sum of all values in a data set to the total number of items in the data set. The mean (average) of a data set is calculated by adding all the numbers and dividing the sum by the number of values in the set. For example, the mean of the first three even numbers is (2+4+6)/3, which is equal to 4. The most frequently and commonly used type of mean is the arithmetic mean. It represents a typical value of a data set. The other types of means are geometric mean, harmonic mean, and weighted mean. If the type of mean is not mentioned, then we take it as the arithmetic mean.
Median Definition
As the name suggests, the median is the middlemost value in a given data set. Before finding the median the data set is arranged in ascending order. Now the middlemost value is considered as the median. If the total number of items in the list is odd, then after arranging the elements in the ascending order the middlemost value is taken as the median. If the number of items in the data set is even, then the average of the two middle values is taken, that is, [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2, where 'n' is the total number of observations. If there are an odd number of items in the data set, then the median is given by the [(n+1)/2]^{th} term, where 'n' is the total number of observations.
What is the Difference Between Mean and Median?
Sometimes there is a need to understand and interpret what a given set of data collected speaks about. In all the cases, we may not be able to use the mean or the median to find the central measure or a value that typically represents a data set. If the list of data contains extremely high and low values, then finding the mean will not give a clear picture of what most of the values in the data set represent. In such cases, the median is useful where the values are arranged in ascending order before finding the central value. An important term to be known here is the outlier. Outliers are the values in a data set that lie completely out of the range of most of the values. For example, in the following set of data: (2, 486, 490, 496, 998); 2 and 998 are outliers since they completely lie out of the range of the other values. While calculating mean and median, it is very important to note if there are any outlier values in the given data set, as it hugely impacts the values of the mean and median.The key differences between mean and median are tabulated below.
Mean  Median 
Mean is the ratio of the sum of all the values of the data set to the total number of values.  Median represents the middlemost value of a data set. 
The data set items are not arranged in ascending order before finding the mean.  The data set items are arranged in ascending order before finding the median. 
The formula to calculate the mean is the same if there are an even or odd number of items in the data set.  The formula is different for an even and an odd number of items in the data set. 
The formula to calculate the arithmetic mean for 'n' observations is the ratio of the sum of all the observations to the total number of observations.  The formula to calculate median: For odd number of observations: [(n+1)/2]^{th} term. For even number of observations: [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2, where 'n' is the total number of observations. 
Mean takes every value in the given data set for the calculation.  Every value in the given data set is not taken while calculating the median. 
Mean is not suitable if there are drastically high and low observations in the data set. This data may not clearly represent the data taken into consideration.  Median is suitable if there are very high and low values since it arranges all the values in ascending order. 
Mean is best suited for the normal distribution of data.  The median value is more appropriate for skewed distribution. 
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Solved Examples

Example 1: Find the mean and median of the following data set.
20, 25, 30, 32, 35,18, 47, 52, 36, 32
Solution:Mean = Sum of all the observations / Total number of observations.
= 20+25+30+32+35+18+47+52+36+32 / 10
= 327/10
= 32.7
Median is the middlemost value. To find the median we first arrange all the numbers in ascending order. So, the list of values after arranging in ascending order are:18, 20, 25, 30, 32, 32, 35, 36, 47, 52.
Since the number of observations are even, the formula to calculate median is
Median = [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2
Here n = 10.
Median = [(10/2)^{th} term + ((10/2) + 1)^{th} term] / 2
= [(5)^{th} term + (5 + 1)^{th} term] / 2
= (32 + 32) / 2
= 64/2
= 32
Therefore, the mean = 32.7 and the median = 32. 
Example 2: For the following data set, find out which measure of central tendency, the mean or median is more appropriate and justify your answer.
1, 25, 28, 32, 36, 40, 42, 345
Solution:
In the given data set, we can clearly observe that there are two values that lie completely out of range of the other values, which are 1 and 345. In such cases, it is better to choose the median to find the central value. Let us find both the mean and the median and see how there is a huge variation.Mean of the given data set = Sum of all the observations / Total number of observations.
= (1+25+28+32+36+40+42+345) / 8
= 68.625
Median is the middlemost value of the given data. The values are already arranged in an ascending order. Since the total number of observations is even, we use the following formula:
Median = [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2
= [(8/2)^{th} term + ((8/2) + 1)^{th} term] / 2
= [(4)^{th} term + ((4) + 1)^{th} term] / 2
= (32 + 36) / 2
= 68/2
= 34
We clearly see that the value of median lies at the center of the set of values in the data set, whereas the mean value does not clearly give the center value. Therefore, for this data set we can prefer to use the median to find the central value.
FAQs on Difference Between Mean and Median
Is There Any Difference Between Mean and Median Values for the First 10 Natural Numbers?
The average of the first ten natural numbers is (1+2+3+4+5+6+7+8+9+10)/10, which is 55/10 or 5.5. Since there is an even number of observations here, we use the following median formula to calculate the median: [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2, where 'n' is the number of observations. (n/2)^{th} term is 5 and ((n/2) + 1)^{th} term is 6. Therefore, the median is (5 + 6)/2, which is 11/2 or 5.5. Hence, the mean and the median values are the same for the first ten natural numbers.
List Out All the Differences Between a Mean and a Median.
Mean and median are different measures of central tendency and are used in different scenarios. The differences between a mean and a median are as follows.
 A mean is the average of the set of values in a given data set, whereas, the median is the central value in a data set.
 The mean is calculated by taking all the values in the data set, whereas, the median value is calculated by taking the [(n+1)/2]^{th} term for an odd number of observations and [(n/2)^{th} term + ((n/2) + 1)^{th} term] / 2 for an even number of observations.
 The data set values need not be arranged in any order before calculating the mean, whereas, the data set values are to be arranged in ascending order before calculating the median.
 Mean is the best measure to use when there are no outliers (which means extreme low and high values), whereas the median is the best measure to use when there are outliers in the given data set.
Will the Mean and the Median Give the Same Range of Values For the Same Set of Values?
No, the mean and the median will not give the same value, because, if there are extremely high and low values in a data set, then the values of mean and median completely lie in a different range. So it is always best to choose the correct type of measures of central tendency before we proceed to find the central value.
Which Measure Can be Used to Find the Average of First Five Odd Numbers? Is it the Mean or the Median?
Since the first five odd numbers lie well in a range up to 10, it makes sense to use either a mean or a median since both the measures give the same value. The mean or the average of the first five odd numbers is 5. The median value is also 5 since it is the middlemost value.
What is an Outlier? Does it Make Any Difference in the Values of Mean and Median?
Outliers are the values in the data set that are completely out of range. For example, in the data set: (1,200,210,215,200,1000), the values 1 and 1000 are completely out of range. They are called outliers. When mean and median are calculated using these values, we cannot get a clear picture of what the other data values are trying to interpret. So, an outlier definitely makes a huge difference in the values of mean and median.
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