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Mean and Median are two measures of central tendency in Statistics along with Mode. Central Tendency is defined as a statistical measure that is used to represent the single value of an entire distribution or dataset.
- Mean is the mathematical average or a calculated central value of a set of numbers.
- Median is the middlemost number or central value in a sorted dataset.
- Mean is an arithmetic average while Median is a positional average.
Difference between Mean and Median lies in the fact that the mean is the average of a set of data while the median is the middle value of the arranged set of data. Mean is the best measure to calculate central tendency in symmetric distributions while the median is the best measure to calculate central tendency in skewed distributions.
Read More: NCERT Solutions For Class 11 Maths Statistics
Key Terms: Mean, Median, Mean Formula, Median Formula, Arithmetic Mean, Geometric Mean, Harmonic Mean, Statistics
What is Mean?
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Mean is the arithmetic average of a given set of data. It is one of the most frequently used measures of central tendency in Statistics.
- Mean is calculated by dividing the sum of all values in a data set by the total number of values in the data set.
- It is commonly referred to as "average".
- It is denoted by the symbol ‘x̄’.
- Mean plays a significant role in finance and is used in various financial fields and business valuation.
Mean Formula is given as:
| x̄ = (x1 + x2 + … + xn)/n |
Where,
- x1, x2, … , xn are the data values.
- n is the total number of values.
Mean, Median and Mode Video Explanation
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Types of Mean
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Mean can be classified into three major types as follows:
- Arithmetic Mean
- Geometric Mean
- Harmonic Mean
All these three types of mean are calculated using different formulas in Statistics depending on the situation or problem provided.
Arithmetic Mean (AM)
Arithmetic Mean is commonly referred to as mean only or arithmetic average. Arithmetic Mean is the sum of all observations divided by the number of observations.
Arithmetic Mean Formula is given as:
| x̄ = Σxi/n |
Where
- xi = Data Values
- n = Total Number of Data Values
Solved ExampleExample: Find the mean of the given data values: 3, 5, 8, 6. Solution: Given that,
Using Arithmetic Mean Formula, x̄ = Σxi/n x̄ = (3+5+8+6)/4 = 22/4 = 5.5 Thus, the arithmetic mean of 3, 5, 8, and 6 is 5.5. |
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Geometric Mean (GM)
Geometric Mean is the average value obtained by taking the nth root of the product of n values in the given set of data. It is used for describing proportional growth.
Geometric Mean Formula is given as:
| G.M. = n√(x1 · x2 · x3 · … · xn) |
Where
- x1, x2, x3, … , xn = Data Values
- n = Total Number of Data Values
Solved ExampleExample: Find the geometric mean of the given data values: 8, 9, 3. Solution: Given that,
Using Geometric Mean Formula, GM = 3√(8 x 9 x 3) = 3√(216) = 6 Thus, the geometric mean of 8, 9, and 3 is 6. |
Harmonic Mean (HM)
Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals. Harmonic Mean Formula is given as:
| H.M. = 1/[Σ(1/xi)]/N = N/Σ(1/xi) |
Where
- xi = Data Values
- N = Total Number of Data Values
Solved ExampleExample: Find the harmonic mean of the given data values: 2, 4, 8. Solution: Given data values are 2, 4, and 8. Thus,
a1 + a1 + a3= 1/2 +1/4 +1/8 = 7/8 Using Harmonic Mean Formula, HM = 1/[Σ(1/xi)]/N = N/Σ(1/xi) HM = 3/(7/8) = 24/7 Thus, the harmonic mean of 2, 4, and 8 is 24/7. |
What is Median?
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Median is one of the three measures of central tendency along with Mean and Mode.
- Median is the value of the middle-most observation obtained after arranging the data in ascending order.
- It is defined as a point at which half the data is more and half the data is less.
- It is a Positional Average as the data placed in the middle of a sequence is taken as the median.
- Median is considered to be the easiest statistical measure to calculate.
Median Formula
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Median is calculated in two different manners depending on the total number of data values.
- Median When n is Odd
- Median When n is Even
Median Formula When n is Odd
If the total number of values is an odd number, then Median Formula is
| Median = [(n + 1)/2]th Term |
Here, n refers to the total number of values.
Solved ExampleExample: Find the median of the given data set: 42, 40, 50, 60, 35, 58, 32. Solution: The given data values are 42, 40, 50, 60, 35, 58, and 32. First, arrange the data values in ascending order as 32, 35, 40, 42, 50, 58, 60. Here, n = 7 (Odd Number). Thus, Using Median Formula When n is Odd, Median = [(n + 1)/2]th term = (7 + 1)/2th term = 4th term = 42 Thus, the median of 42, 40, 50, 60, 35, 58, and 32 is 42. |
Median Formula When n is Even
If the total number of values is an even number, then Median Formula is
| Median = [(n/2)th Term + ((n/2) + 1)th Term]/2 |
Here, n refers to the total number of values.
Solved ExampleExample: Find the median of the given data set: 142, 140, 130, 150, 160, 135, 158, 132. Solution: The given data values are 142, 140, 130, 150, 160, 135, 158, and 132. Arrange the values in ascending order as 130, 132, 135, 140, 142, 150, 158, 160. Here, n = 8 (Even Number). Thus, Using Median Formula When n is Even, Median = [(n/2)th term + ((n/2) + 1)th term]/2 Median = [(8/2)th term + ((8/2) + 1)th term]/2 = (4th term + 5th term)/2 = (140 + 142)/2 = 141 Thus, the median of 142, 140, 130, 150, 160, 135, 158, and 132 is 141. |
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Difference between Mean and Median
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Mean and Median are both measures of central tendency used in Statistics.
- Mean is defined as the average of all the values in a data set.
- Median is the middle value of an arranged data set.
Difference between Mean and Median is given as follows:
| Mean | Median |
|---|---|
| Mean is the average value of a given set of data. | Median is the middlemost term of an arranged set of data. |
| Mean is an Arithmetic Average. | Median is a Positional Average. |
| It is used for symmetric distributions and normal data. | It is used for skewed distributions. |
| It is calculated by adding all the values and dividing the total by the number of values. | It is calculated by arranging the numbers in order and then finding the number in the center of the distribution. |
| Mean Formula is the same for an even and odd number of items in the data set. | Median Formula is given separately for an even and an odd number of items in the data set. |
| Mean is sensitive to outlier data. | Median is not sensitive to outlier data. |
| It considers every value of data. | It does not consider every value in the data. |
| Mean gives the central value or center of gravity of the data. | Median gives the midpoint of the data. |
| External factors limit the application of the mean. | It is reliable for uneven data as well. |
| Extreme values affect the value of the mean. | Median remains unaffected by extreme values. |
Things to Remember
- Mean and Median are the two most popular measures of central tendency.
- Mean is the arithmetic average of a given set of data.
- Mean Formula is given as Mean = (x1+ x2+ … + xn)/n.
- Mean is classified into three major types namely Arithmetic Mean, Geometric Mean, and Harmonic Mean.
- Median is the middlemost number or value in a set of arranged data.
- It divides the set of data into the lower half and the upper half.
- Median Formula for n (odd) number of terms is M = [(n+1)/2]th Term.
- Median Formula for n (even) number of terms is M = [(n/2)th Term + ((n/2) + 1)th Term]/2.
- Median is a positional average while Mean is an arithmetic average.
- Extreme values affect mean while they do not affect the median.
- Mean is used for symmetric distributions while Median is used for skewed distributions.
Previous Years’ Questions
- The mean and variance of 20 observations are found to be 10 and 4... (JEE Main - 2020)
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- If coefficient of variation is 60 and standard deviation is 24, then Arithmetic mean... (KCET - 2017)
- The arithmetic mean of the data 0,1,2,……,n with frequencies... (BITSAT – 2015)
- The mean of the data set comprising of 16 observations is 16. If one of the... (JEE Main - 2016)
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Sample Questions
Ques. Find the mean and median of the data 8, 10, 14, 17, 26. (3 Marks)
Ans. Given data values are 8, 10, 14, 17, and 26.
Using the Mean Formula,
Mean = Sum of All Observation/Total Number of Observations
Mean = (8+10+14+17+26)/5
= 75/5 = 17
For the median, first arrange the data in ascending order.
8, 10, 14, 17, 26
Since the number of terms is odd, Using the Median Formula When n is Odd,
Median = [(n + 1)/2]th term = (5 + 1)/2 = 6/2 = 3rd Term = 14
Thus, the mean and median of the values 8, 10, 14, 17, and 26 are 17 and 14 respectively.
Ques. Find the median for the data: 3, 3, 4, 6, 6, and 14. (3 Marks)
Ans. Given data values are 3, 3, 4, 6, 6, and 14.
Arrange the values in ascending order as 3, 3, 4, 6, 6, and 14.
Here, n = 8 (Even Number).
Thus, Using Median Formula When n is Even,
Median = [(n/2)th term + ((n/2) + 1)th term]/2
Median = [(6/2)th term + ((6/2) + 1)th term]/2
= (3th term + 4th term)/2 = (4 + 6)/2 = 10/2 = 5
Thus, the median of 3, 3, 4, 6, 6, and 14 is 5.
Ques. The mean of 5 numbers is 18. If a number is excluded from the data set, the mean is 16. Find the excluded number. (3 Marks)
Ans. Mean Formula is given as:
Mean = Sum of All Observation/Total Number of Observations
Given, the mean is 18 when the number of terms is 5, and the mean is 16 when the number of terms is 4. Thus,
18 = Sum of 5 Terms/5
Sum of 5 Terms = 18 x 5 = 90…..(1)
16 = Sum of 4 Terms/4
Sum of 4 Terms = 16 x 4 = 64…..(2)
To find the excluded number, subtract the sum of 4 terms from the sum of 5 terms.
Excluded Number = 90 - 64 = 26
Thus, the excluded number is 26.
Ques. State the major differences between Mean and Median. (3 Marks)
Ans. The major differences between Mean and Median are as follows:
- Mean is the average of the set of values in a given data set, while the median is the central value in a data set.
- The data values need not be arranged in any order for calculating the mean, while the data values are to be arranged in ascending order to calculate the median.
- Mean is majorly used when there are no extremely low and high values, whereas the median is the best measure to use when there are extreme values.
Ques. A team plays 4 matches. The goals scored in each match are 6, 7, 4, and 7. How many goals do they need to score in their fifth match to raise their average to 7 goals per match? (3 Marks)
Ans. Let x be the goal scored in the fifth match.
Given,
- Number of Terms = 5
- Mean= 7
Sum of Terms = 6 + 7 + 4 + 7 + x
Using the Mean Formula,
Mean = Sum of All Observation/Total Number of Observations
7= (6+7+4+7+x)/5
35 = 6+7+4+7+x
35 = 24+x
x = 35-24 = 11
Thus, the players need to score 11 goals in their fifth match to raise their average to 7 goals per match.
Ques. Find the mean of the first five whole numbers. (2 Marks)
Ans. Given data values are 0, 1, 2, 3, and 4.
Using the Mean Formula,
Mean = Sum of All Observation/Total Number of Observations
Mean = (0+1+2+3+4)/5
= 10/5 = 2
Thus, the mean of the first five whole numbers is 2.
Ques. What are the mean and median of the given data values: 20, 25, 30, 32, 35,18, 47, 52, 36, 32? (5 Marks)
Ans. Given data values are 20, 25, 30, 32, 35,18, 47, 52, 36, and 32.
Using the Mean Formula,
Mean = Sum of All Observation/Total Number of Observations
= 20+25+30+32+35+18+47+52+36+32 / 10
= 327/10 = 32.7
To calculate the median, arrange the data values in ascending order.
18, 20, 25, 30, 32, 32, 35, 36, 47, 52.
As n = 10 (Even Number).
Thus, Using Median Formula When n is Even,
Median = [(n/2)th term + ((n/2) + 1)th term]/2
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
Thus, the mean and median of the given data values are 32.7 and 32 respectively.
Ques. Find the median of 3, 15, 2, 34, 11, and 25. (3 Marks)
Ans. Arranging the data values in ascending order, we get
2, 3, 11, 15, 25, 34
As n = 6 (Even Number).
Using Median Formula When n is Even,
Median = [(n/2)th term + ((n/2) + 1)th term]/2
Median = [(6/2)rd term + ((6/2)th term + 1)]/2 = (3rd term + 4th term)/2
= (11 + 15)/2 = 26/2 = 13
Thus, the median of the given data values is 13.
Ques. Calculate the mean age of the group of 10 people with the following ages: 45, 39, 53, 45, 43, 48, 50, 40, 40, 45. (3 Marks)
Ans. Given that,
n = 10
x1 = 45, x2 = 39, x3 = 53, … , xn = 45
Using the Mean Formula,
x̄ = (x1+ x2+ x3+ … + xn )/n
= (45 + 39 + 53 + 45 + 43 + 48 + 50 + 40 + 40 + 45)/10 = 448/10 = 44.8
Thus, the mean age of the group is 44.8.
Ques. What are the types of Mean? (2 Marks)
Ans. There are three types of Mean which are as follows:
- Arithmetic Mean
- Geometric Mean
- Harmonic Mean
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