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Updated on 15th May, 2023 , 7 min read
Understanding the mean median mode relation is crucial for dealing with similar issues in statistics. In the case of a moderately skewed distribution, or generally speaking, the difference between the mean and median is equal to three times the difference between the mean and median.
Mean has been learned by most of the people in middle school math as the norm. The mean is what you get by combining all the values and dividing the total by the number of values, given a set of values. You have a set of values called the population X1,......,Xnwritten in math notation.
The mean number is very helpful. It describes the group's property. It is important to understand that the mean is not an entity-in reality, there can be anything whose value matches the mean, but the mean is a summarized representation of the population.
Example: Calculate the average value of the following numbers: 6, 7, and 5.
Solution: To calculate the value of Mean,
The given observations' total is equal to 6 + 4 + 2 = 18.
Therefore, we must divide the sum by the total number of values in order to determine the mean value.
3 values in total make up the question in this case.
Consequently, Mean Value = 18 / 3 = 6
Formula |
Mean = sum of the terms / number of terms |
The median frequently serves as a better proxy for the average group member. The median will be the number in the middle of a list if all the values are taken and arranged in ascending order. A characteristic that all group members share is the median. The mean might not be particularly close to the quality of any group member, according to the value distribution. The mean is also subject to skewing; even one value that differs noticeably from the rest of the group can significantly alter the mean. The median gives you a central group member without the skew effect that outliers introduce. The median value in a normal distribution represents a representative sample of the population.
Example: Calculate the median between the numbers 42, 16, and 36.
Solution:The number 16 is the median of the numbers 42, 16, and 36 because it falls in the middle when the numbers are written as (42, 16, 36). and, Median helps determine the middle value in an observation.
Formula |
Formula- median = [(n+1)/2]th (If n is Odd) Or Median = [(n/2)th + {(n/2) +1}th]/2 (if n is even) |
The group's most prevalent member is the mode. It doesn't matter which value is the most prevalent; the mode is always the largest or smallest in the group. Because they are typically the least meaningful, the majority of these three median measures are also the least used. However, it can be useful occasionally. The median, mean, and mode will all be the same if your data is both regular and flawless. This is practically unimaginable and never occurs in real life.
Example: What is the given observation's mode value? {2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4, 4}
Solution: The given observation's mode value is 4. Its values are 2, 1, 4, 4, 4, 6, 8, 4, 1, 8, 4, 4, 9, 4. This is due to the fact that 4 occurred the most frequently during the course of the observation.
Formula |
Mode = Most number of times occurring value in the data |
The equation 3 (median) = mode + 2 mean can be used to define the mean median mode relation in a moderately skewed distribution. Karl Pearson developed the following formula, which can be used to understand the proof of the mean, median, and mode formula:
(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
When dealing with a distribution that is moderately skewed, the difference between the mean and mode can be approximated as being three times the difference between the mean and median. This relationship between the empirical mean, median, and mode can be expressed as:
Mean – Mode = 3 (Mean – Median)
Or
Mode = 3 Median – 2 Mean
Either of these two ways of expressing equations can be used as per convenience since, by expanding the first representation, we get the second one as shown below:
Mean – Mode = 3 (Mean – Median)
Mean – Mode = 3 Mean – 3 Median
By rearranging the terms,
Mode = Mean – 3 Mean + 3 Median
Mode = 3 Median – 2 Mean
However, we can define the relation between mean, median and mode for different types of distributions, as explained below:
When a frequency distribution graph displays a symmetrical frequency curve, the values of the mean, median, and mode will be identical.
Mean = Median = Mode
For a frequency distribution that is positively skewed, the mean will always be greater than the median, and the median will always be greater than the mode.
Mean > Median > Mode
If a frequency distribution has a negative skew, then the mean will always be less than the median, and the median will always be less than the mode.
Mean < Median < Mode
The following table has all the information about the difference between the mean, mode, and median on the following basis:
Basis |
Mean |
Median |
Mode |
Meaning |
The arithmetic mean is calculated by dividing the total number of observations by the number of items is used to calculate the average. |
The median is the value that the same number of observations both exceed and fall short of. |
The value of the observations that occur the most frequently makes up the set's mode. |
Formulae |
Mean is the product of the observational total and the observational sum. |
The following formula is used if n is odd: (n+1/2th observation) value Value of (n/2)th observation plus (n/2+1)th observation divided by two is used if n is even. |
Value that occurs the most frequently in the data. Mean = 2 - 3 Median |
Preference |
When the data is normally distributed, the mean is preferred. |
When the data distribution is erratic, the median is preferred. |
When the distribution of the data is nominal, the mode is preferred. |
Calculate |
Divide the total number of observations by the sum of all the data's numbers. |
Sort the data in ascending and descending order, determine how many observations there were, and then use the formula as necessary. |
The value of the observations that appear the most frequently make up the mode of a set of observations. |
Do you struggle to understand the mean, median, and mode differences? Here are a few pointers that might be useful.
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SGPA Calculator |
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SGPA to CGPA |
Question 1: The mean is 22.5 and the median is 20, in a moderately skewed distribution. Find the mode's approximation using these values.
Solution: Given,
Mean = 22.5
Median = 20
Mode = x
Using the correlation between mean, mode, and median that we have discovered,
(Mean – Mode) = 3 (Mean – Median)
So,
22.5 – x = 3 (22.5 – 20)
22.5 – x = 7.5
∴x = 15
So, Mode = 15.
Question 2: In a distribution that is moderately skewed, the median value is 10 and the mean value is 12. Find the approximate value of the mode using these values.
Solution: We know that in a moderately skewed distribution, the relationship between the mean, median, and mode is 3 median = mode + 2 mean. Let's assume that mode is "x." It has been stated that the mean is 12 and the median is 10. 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.
Question 3: If the mean and mode are 30 and 20, respectively, then determine the potential range of the median for a positively skewed distribution.
Solution: The empirical relationship between the mean, median, and mode for a frequency distribution that is positively skewed is mean > median > mode.Based on this, the range of the median is 30 > median > 20 if the mean is 30 and the mode is 20. The median will therefore be higher than 20 and lower than 30.
Question 4: If the mode is 35.3 and the mean is 30.5, then use an empirical formula to find the median of the data.
Answer: 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
Question 5: Given that the mean and median of a moderately skewed distribution are 10 and 12, respectively. Find the mode's approximation using these values.
Answer: We are aware that in a moderately skewed distribution, the relationship between the mean, median, and mode is 3 median = 2 mode + 2 mean. Let's assume that "x" is the mode. It has been stated that the mean is 12 and the median is 10. Now that we have the mean, mode, and median relationship, we can:
3 × 10 = x + 2 × 12
30 = x + 24
x = 30-24
x = 6
Therefore, the value of mode is 6.
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By - Nikita Parmar 2024-09-06 10:59:22 , 6 min readAns. Numbers that represent an entire collection of data or information include mean, median, and mode. The measures of central tendency are collectively referred to as mean, median, and mode.
Ans. The "empirical relationship," which is defined as Mode being equal to the difference between 3 times the median and 2 times the mean, is the name given to this relationship between mean, median, and mode.
Ans. When every observed value occurs exactly once in a data set, there is no mode.
Ans. Mode: The most common number, or the number that appears the most frequently. Example: Since the number 2 appears three times, more than any other number, it is the mode of the numbers 4, 2, 4, 3, and 2.
Ans. The letters Mo or Mode are used to represent it.