What are the relationships between the mode median and mean when in a skewed distribution?

Answer - Relationship between Mean Median and Mode is known as the “Empirical Relationship”.

The mean of a data set is determined by adding together all the data values and dividing the result by the total number of data sets. By placing the values in either ascending or descending order and then selecting the middle value, the median, the middle value among the observed set of values is determined. By counting the occurrences of each data value, the mode from a data collection with the highest frequency is determined.

When the distribution is substantially skewed, the gap between the mean and the mode is almost three times that between the mean and the median. Consequently, the following is the empirical mean median mode relation:

Mean – Mode = 3 (Mean – Median)

Or

Mode = 3 Median – 2 Mean

Summary:

Relationship between Mean Median and Mode

The relationship between Mean Median and Mode is known as an “empirical relationship”. It is characterized as being the difference between the median and the mean times three.

Answer

Verified

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:

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.

Consider the following data set.
4; 5; 6; 6; 6; 7; 7; 7; 7; 7; 7; 8; 8; 8; 9; 10

This data set can be represented by following histogram. Each interval has width one, and each value is located in the middle of an interval.

Figure 2.11

The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. 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.

The histogram for the data: 4; 5; 6; 6; 6; 7; 7; 7; 7; 8 shown in Figure 2.11 is not symmetrical. The right-hand side seems "chopped off" compared to the left side. A distribution of this type is called skewed to the left because it is pulled out to the left. We can formally measure the skewness of a distribution just as we can mathematically measure the center weight of the data or its general "speadness". The mathematical formula for skewness is: a3=∑(xi−x¯)3ns3a 3=∑(xi−x¯)3ns3. The greater the deviation from zero indicates a greater degree of skewness. If the skewness is negative then the distribution is skewed left as in Figure 2.12. A positive measure of skewness indicates right skewness such as Figure 2.13.

Figure 2.12

The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so.

The histogram for the data: 6; 7; 7; 7; 7; 8; 8; 8; 9; 10 shown in Figure 2.13, is also not symmetrical. It is skewed to the right.

Figure 2.13

The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most.

The mean is affected by outliers that do not influence the mean. Therefore, when the distribution of data is skewed to the left, the mean is often less than the median. When the distribution is skewed to the right, the mean is often greater than the median. In symmetric distributions, we expect the mean and median to be approximately equal in value. This is an important connection between the shape of the distribution and the relationship of the mean and median. It is not, however, true for every data set. The most common exceptions occur in sets of discrete data.

As with the mean, median and mode, and as we will see shortly, the variance, there are mathematical formulas that give us precise measures of these characteristics of the distribution of the data. Again looking at the formula for skewness we see that this is a relationship between the mean of the data and the individual observations cubed.

a3=∑(xi−x ¯)3ns3a3=∑(xi−x¯)3ns3

where ss is the sample standard deviation of the data, XiXi , and x¯ x¯ is the arithmetic mean and nn is the sample size.

Formally the arithmetic mean is known as the first moment of the distribution. The second moment we will see is the variance, and skewness is the third moment. The variance measures the squared differences of the data from the mean and skewness measures the cubed differences of the data from the mean. While a variance can never be a negative number, the measure of skewness can and this is how we determine if the data are skewed right of left. The skewness for a normal distribution is zero, and any symmetric data should have skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. The skewness characterizes the degree of asymmetry of a distribution around its mean. While the mean and standard deviation are dimensional quantities (this is why we will take the square root of the variance ) that is, have the same units as the measured quantities XiXi, the skewness is conventionally defined in such a way as to make it nondimensional. It is a pure number that characterizes only the shape of the distribution. A positive value of skewness signifies a distribution with an asymmetric tail extending out towards more positive X and a negative value signifies a distribution whose tail extends out towards more negative X. A zero measure of skewness will indicate a symmetrical distribution.

Skewness and symmetry become important when we discuss probability distributions in later chapters.

What is the relationship between mean median and mode for a skewed distribution?

What is the relationship between the mean median and mode of a normal distribution?

How does the mean median and mode scores indicate if the distribution is normal or skewed?

Which of the following describes the relationship between mean median and mode for a moderately skewed distribution?