What is the relationship among the mean, median, and mode in a symmetric distribution?

Q: What is the relationship between the mean, median and mode in a symmetric distribution?

In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.

Q: What is the relationship among the mean, median and mode in a symmetric distribution select one?

They are all equal . The value of mean, median and mode all are equal in a symmetric distribution because all 3 values occur at the same point as the values of the variables appear at the regular frequncies. When there is symmetrical distribution mean median and mode lies in the middle of the graph.

Q: What is the relationship between mean, median and mode in asymmetrical distribution?

The relation between mean, median and mode for an asymmetrical distribution is given by: Mode = 3 Median - 2 Mean.

Q: Is mean, median mode in symmetric distribution?

In symmetrical distribution, Mean=Mode=Median.

The relationship between mean, median and mode for a moderately skewed distribution isA. $\text{Mode=2Median-3Mean}$B. $\text{Mode=Median+2Mean}$C. $\text{Mode=3Median-Mean}$D. $\text{Mode=3Median-2Mean}$

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The relationship between mean, median and mode for a moderately skewed distribution isA. $\text{Mode=2Median-3Mean}$B. $\text{Mode=Median+2Mean}$C. $\text{Mode=3Median-Mean}$D. $\text{Mode=3Median-2Mean}$
What is the relationship between the mean, median and mode in a symmetric distribution?
What is the relationship among the mean, median and mode in a symmetric distribution select one?
What is the relationship between mean, median and mode in asymmetrical distribution?
Is mean, median mode in symmetric distribution?

Answer

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Hint: To find the relationship between mean, median and mode for a moderately skewed distribution, we should express this relationship by Karl Pearson’s formula. It is defined as the distance between the mean and the median is about one-third the distance between the mean and the mode. We can write this as \[\text{Mean}-\text{Median=}\dfrac{1}{3}\text{ }\left( \text{Mean}-\text{Mode} \right)\] . When solving this, we will get the required solution.

Complete step-by-step answer:
We need to find the relationship between mean, median and mode for a moderately skewed distribution. Let us see what skewed distribution is.

The above figure shows normal distribution. ‘A’ is known as the tail. In this distribution, we can see that $\text{Mean}=\text{Median}=\text{Mode}$ . If one tail is longer than another, the distribution becomes skewed. These are sometimes called asymmetric or asymmetrical distributions.
Let us see the types of skew distribution. There are 2 types –positive and negative.
Negatively-skewed distributions have a long left tail. They are also called negatively-skewed distributions. We can represent it as follows:

In the above figure, green colour indicates mode, red is median and violet is mean. From the figure, we can get the following relation
$\text{Mode}>\text{Median}>\text{Mean}$
Now, let us see the positively-skewed distribution.
Positive-skew distributions have a long right tail. They are also called right-skewed distributions. We can represent it as follows:

From the above figure, we can observe that
$\text{Mode}<\text{Median}<\text{Mean}$
Now, let us see what moderately-skewed distribution is.
Highly skewed: Skewness is less than -1 or greater than 1
Moderately skewed: Skewness between -1 and -0.5 or between 0.5 and 1.
Approximately symmetric: Skewness is between -0.5 and 0.5.
We can express the relationship between mean, median and mode by Karl Pearson’s formula. It is defined as the distance between the mean and the median is about one-third the distance between the mean and the mode.
\[\text{Mean}-\text{Median=}\dfrac{1}{3}\text{ }\left( \text{Mean}-\text{Mode} \right)\]
Let us now solve this. We will get
\[\text{Mode}=\text{Mean}-3\left( \text{Mean}-\text{Median} \right)\]
Let us now simplify the RHS. We will get
\[\text{Mode}=\text{Mean}-3\text{Mean+}3\text{Median}\]
\[\begin{align}
& \Rightarrow \text{Mode}=-2\text{Mean+}3\text{Median} \\
& \Rightarrow \text{Mode}=3\text{Median}-2\text{Mean} \\
\end{align}\]

So, the correct answer is “Option D”.

Note: You may make an error when writing the formula \[\text{Mean}-\text{Median=}\dfrac{1}{3}\text{ }\left( \text{Mean}-\text{Mode} \right)\] as \[\text{Mean}-\text{Mode=}\dfrac{1}{3}\text{ }\left( \text{Mean}-\text{Median} \right)\] . Do all the calculations carefully, else the required solution will not be reached. The relation \[\text{Mode}=3\text{Median}-2\text{Mean}\] is used for moderate skewed distribution only.

After reading this article you will learn about the relationship between mean, median and mode.

In the case of unimodal distributions, relationships may be stated as under:

(i) in the case of a perfectly symmetrical distribution, mean, median and mode are equal. The relationship is also shown in fig 3.3.

(ii) For moderately asymmetrical distributions, the locations of mean, median and mode are shown in Figure 3.4. In the ease of positively skewed curve, the mean shall have the highest value, the mode the lowest and the median will be about one-third the distance from the mean towards the mode.

On the other hand, for a negatively skewed curve, the mean will be the lowest, the mode the largest and median will still be approximately at a one-third distance from mean towards the mode.

These empirical relationships for moderately asymmetrical distribution may be put in the form of formula which reads:

What is the relationship between the mean, median and mode in a symmetric distribution?

In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.