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Relation between Mean, Median, and Mode is that Mode is equal to the difference between 3 times median and 2 times mean. It is expressed using Karl Pearson's formula.
- The mean or average of a data set is found by adding all numbers in the data set and dividing them by the total number of values.
- The Median is the middle value when a data set is arranged from the least to the greatest.
- Mode is the number that occurs most frequently in a data set.
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Key Terms: Mean, Median, Mode, Median Formula, Central tendency, Median of grouped data, Relation between Mean, Median, Mode, Karl Pearson’s Formula.
Empirical Relationship Between Mean, Median & Mode
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In statistics, a relation exists between mean, median, and mode for moderately skewed distribution.
- This mean median and mode relationship is known as the “empirical relationship” which is defined as Mode that is equal to the difference between 3 times the median and 2 times the mean.
- The proof of the mean, median, and mode formula can be understood using Karl Pearson’s Formula, which is given below:
(Mean - Median) = 1/3 (Mean - Mode)
3 (Mean - Median) = (Mean - Mode)
3 Mean - 3 Median = Mean - Mode
3 Median = 3 Mean - Mean + Mode
| 3 Median = 2 Mean + Mode |
Mean, Median and Mode Video Lecture
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|---|---|---|
| Interquartile range | Statistical Inference | Mode of Grouped Data |
| Absolute Values | Sampling Error Formula | Mean of Grouped Data |
Example of Relation between Mean, Median, and ModeQues. In a moderately skewed distribution, the median is 40 and the mean is 42. Using these values, find the approximate value of the mode. (3 Marks) Ans. Given, Mean = 42 Median = 40 Mode = x Now, using the relationship between mean mode and median we get, (Mean – Mode) = 3 (Mean – Median) So, 42 – x = 3 (42 – 40) 42 – x = 6 ∴ x = 36 So, Mode = 36 |
Frequency Distribution & Mean, Median, Mode
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The frequency curves and the relationship between mean, median and mode for different types of distributions can be demonstrated as below.
Frequency Distribution with Symmetrical Frequency Curve
Mean, median, and mode will be equal in the case of a frequency distribution graph having a symmetrical frequency curve, as shown below:
Mean = Median = Mode
Frequency Distribution with Symmetrical Frequency Curve
For Positively Skewed Frequency Distribution
The mean is always greater than the median and the median is always greater than the mode, if the frequency distribution is positively skewed, as shown below:
Mean > Median > Mode
Positively Skewed Frequency Distribution
For Negatively Skewed Frequency Distribution:
The mean is always lesser than the median and the median is always lesser than the mode, if the frequency distribution is negatively skewed, as shown below:
Mean < Median < Mode
Negatively Skewed Frequency Distribution
Read More:
| Related Articles | ||
|---|---|---|
| Cumulative Frequency Curve | Step Deviation | Frequency Distribution Table |
| Standard Deviation | Dispersion | Variance |
Things to Remember
- The relationship between mean, median, and mode is given by the empirical relation where Mode is equal to the difference between 3 times the median and 2 times the mean.
- The relation between Mean, Median, and Mode is expressed using Karl Pearson's formula.
- If there is an odd amount of numbers, the median value is the number that is in the middle.
- If there is an even amount of numbers, the median is the simple average of the middle pair in the dataset.
- A Median is much more effective than a mean because it eliminates the outliers.
- A set of numbers may have one mode, more than one mode, or no mode at all.
- Mean, median, and mode try to summarize a dataset with a single number to represent a "typical" data point from the dataset.
Previous Year Questions
- If the mean and the variance of a binomial variate… (JEE Main 2015)
- The variance of first n even natural numbers is… (JEE Main 2009)
- A measure of dispersion is indicative of the reliability…
- The mean and variance of 7 observations are 8 and…
- The mean and variance of 20 observations are found to… (JEE Main 2020)
- The variance of the numbers 2, 3, 11 and x is…
- The mean and standard deviation of marks obtained…
- The variance of the first 50 even natural numbers is… (JEE Main 2014)
- The variance of the first 20 natural numbers is…
- The mean of 5 observations is 5 and their variance is 124. If three… (JEE Main 2016)
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Sample Questions
Ques. Find the possible range of the median of a positively skewed distribution, if the mean and mode values are 30 and 20 respectively. (2 Marks)
Ans. For a positively skewed frequency distribution, the empirical relation between mean, median, and mode is mean > median > mode. On the basis of this, the range of the median if the mean is 30 and the mode is 20 is 30 > median > 20. It means that the median will be greater than 20 and less than 30.
Ques. Find the median of the data using an empirical formula, when it is given that mode = 35.3 and mean = 30.5. (2 Marks) [2014]
Ans. Mode = 3(Median) – 2(Mean)
35.3 = 3(Median) – 2(30.5)
35.3 = 3(Median) – 61
96.3 = 3 Median
Median = 96.33 = 32.1
Ques. The mean of the following data is 18.75. Find the value of P: (2 Marks) [2012, 2017D]
Ans.
∴ Mean = ∑fixi/∑fi
18.75/1 = 360+7P/32
360+7P = 600
7P = 600-360 = 240
P = 240/7
P = 34.29 (approx.)
Ques. Following is the age distribution of dengue patients admitted in a hospital during a week of October, 2013: (3 Marks) [2014]
Draw a ‘less than type’ ogive for the above distribution. Also, obtain the median from the curve.
Ans. 
Ques. It is given that in a moderately skewed distribution, median = 10 and mean= 12. Using these values, find the approximate value of the mode. (3 Marks)
Ans. We know that the relation between mean, median, and mode in a moderately skewed distribution is 3 median = mode + 2 mean. Let us take the mode to be ‘x’. We have been given that the median = 10 and mean = 12. Now, using the relationship between mean, mode, and median we get,
3 × 10 = x + 2 × 12
30 = x + 24
x = 30-24
x = 6
Therefore, the value of mode is 6.
Ques. In a moderately skewed distribution, the median is 20 and the mean is 22.5. Using these values, find the approximate value of the mode. (3 Marks)
Ans. Given,
Mean = 22.5
Median = 20
Mode = x
Now, using the relationship between mean mode and median we get,
(Mean – Mode) = 3 (Mean – Median)
So,
22.5 – x = 3 (22.5 – 20)
22.5 – x = 7.5
∴ x = 15
So, Mode = 15.
Ques. Calculate the mean, mode, and median of the salaries of a group of 24 individuals. The values are as given below: (5 Marks)
Rs 20,000; Rs. 20,000; Rs. 22,000; Rs. 25,000; Rs. 25,000; Rs. 28,000; Rs. 30,000; Rs. 31,000; Rs. 32,000; Rs. 34,000; Rs. 35,000; Rs. 37,000; Rs. 39,000; Rs. 40,000; Rs. 42,000; Rs. 43,000; Rs. 80,000; Rs. 1,00,000; Rs. 3,00,000; Rs. 7,00,000; Rs. 30,00,000
Ans. For calculation of the mean,
The sum of the income is Rs.4789000
As there are total of 24 members, the Mean = 20000
This group’s mean income is about Rs.20000.
Now,
We set the values for the median, with half of the values on one side and half on the other. We’re going to write the number of thousands without the trailing number “000” to make it fit:
| 20, 20, 22, 25, 25, 28, 30, 31, 32, 34, 35, 37,39, 39, 40, 42, 42, 43, 80, 100, 300, 700, 3000 |
Median Calculation
Thus, the median is the price on either side of it with the same number of things. In this scenario, we have an even number of members in our community, which implies we should choose one of the two. In this case, as we look at uneven wage distributions, skewing it up and picking the larger of the two will reduce the imbalance, but certainly not eliminate it. Skewing it down will make the difference even bigger than earlier, exaggerating the results.
Thus, picking up the middle value which is higher: Rs.37000.
Mode Calculation
From the definition, the mode in this example is Rs.25000. As value Rs.25000 has fallen in the group most frequently. This case clearly shows the mean’s skewed effect. The mean value is greater than 80% of the group’s actual members. Even if we select the larger of the median’s two possible values, the mean is more than 5 times larger than the median. The Median is a much better measure of the group’s typical member. Mode is not particularly meaningful in this case.
Let’s assume, we had 10 employees and you gave them percentage increase in income as follows:
-2%, -2%, 0%, 0%, 0%, 1%, 3%, 20%
Then median wage change would be +2%. Yet, half of the workers saw either no improvement or a, and in addition, virtually all the raises went to just one person. Take that one user out and the raise goes up by almost 20% to 0.11% on the median.
Ques. Find the approximate value of the mode using these values. Given: that in a moderately skewed distribution, median = 10 and mean = 12. (3 Marks)
Ans. The relationship between mean, median, and mode in a moderately skewed distribution is 3 median = mode + 2 mean.
Let the mode be ‘x’.
Given that the median = 10 and mean = 12.
Now, using the relationship between mean, mode, and median,
3 × 10 = x + 2 × 12
30 = x + 24
x = 30-24
x = 6
Therefore, the value of mode is 6.
Ques. What is Mode? (1 Mark)
Ans. In mathematics, the word "mode" describes the value that appears the most frequently in a particular set. It is, in a nutshell, the number that appears the most frequently in a particular set of data.
Ques. What is Mean? (1 Mark)
Ans. The mathematical concept of "mean" refers to the numerical average of a set of numbers. A set of values can be added up, and the mean can be found by dividing the sum by the total number of values in the set.
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