If mean and mode of the distribution is 32 and 21, then the distribution:
is positively skewed
Understanding Distribution Skewness
Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It indicates whether the data are concentrated more on one side or the other of the mean.
There are three main types of skewness:
- Symmetric Distribution: In a perfectly symmetric distribution (like the normal distribution), the mean, median, and mode are all equal. The distribution looks the same on both sides of the center.
- Positively Skewed Distribution: Also known as right-skewed. In this type, the tail of the distribution extends towards the right. The mean is typically greater than the median, which is typically greater than the mode (Mean > Median > Mode). This happens when there are extreme values on the right side of the distribution, pulling the mean towards the higher values.
- Negatively Skewed Distribution: Also known as left-skewed. In this type, the tail of the distribution extends towards the left. The mean is typically less than the median, which is typically less than the mode (Mean < Median < Mode). This happens when there are extreme values on the left side of the distribution, pulling the mean towards the lower values.
Analyzing the Given Distribution
We are given the following information for the distribution:
- Mean ($ \bar{x} $): 32
- Mode: 21
Now, let's compare the values of the mean and the mode:
$ \text{Mean} = 32 $
$ \text{Mode} = 21 $
Comparing them, we see that $ 32 > 21 $, which means the Mean is greater than the Mode.
Determining Skewness Based on Mean and Mode
Based on the general relationship between the mean and mode in skewed distributions:
- If Mean > Mode, the distribution is likely positively skewed.
- If Mean < Mode, the distribution is likely negatively skewed.
- If Mean $ \approx $ Mode, the distribution is likely symmetric or close to symmetric.
Since our calculated Mean (32) is greater than the given Mode (21), the distribution is indicated to be positively skewed.
Conclusion on Distribution Skewness
Given that the mean of the distribution is 32 and the mode is 21, the mean is greater than the mode. This relationship between the mean and mode is characteristic of a positively skewed distribution.
Revision Table: Skewness Indicators
| Distribution Type | Relationship (Mean, Median, Mode) | Tail Direction |
|---|---|---|
| Symmetric | Mean $ \approx $ Median $ \approx $ Mode | None (balanced) |
| Positively Skewed (Right-Skewed) | Mean > Median > Mode (generally) | Right |
| Negatively Skewed (Left-Skewed) | Mean < Median < Mode (generally) | Left |
Additional Information on Skewness Measurement
While the comparison of mean and mode provides a good indication, skewness can be formally calculated using various coefficients. Common measures include Pearson's coefficient of skewness (Type 1 and Type 2) and the moment coefficient of skewness.
- Pearson's First Coefficient of Skewness: $ Sk_1 = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}} $
- Pearson's Second Coefficient of Skewness: $ Sk_2 = \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}} $
A positive coefficient indicates positive skewness, a negative coefficient indicates negative skewness, and a coefficient near zero indicates symmetry.
Similar Questions
The average of 25 numbers is 36. If the average of first 13 numbers is 32 and that of the last 13 numbers is 39, then the 13th number is
The mode (correct to two decimal places) for the given data is:
Class- interval
Frequency
0-10
6
10-20
9
20-30
8
30-40
14
40-50
28
50-60
20
60-70
11
70-80
9
The arithmetic mean of marks of the students for the given data is:
Marks
No. of students
0-10
12
10-20
18
20-30
27
30-40
20
40-50
17
50-60
60
The incomes of the employees in a state is assumed to be normally distributed with mean Rs.15,000 and variance Rs. 900. The median of the distribution of the income is:
For a moderately skewed distribution, let mode = 15, median = 17.4. The value of mean is:
For the data set with the following observations, the first and second quartiles are:
20, 22, 23, 22, 23, 22, 22, 21, 19, 22, 22, 26, 23, 24, 19, 21, 22, 16
For an asymmetric distribution, if the mean and median are 20 and 30, then the value of the mode is:
For a data set with 24 observations given below, the median is:
10, 11, 13, 13, 18, 20, 22, 22, 24, 24, 25, 29, 30, 31, 35, 37, 37, 37, 46, 51, 54, 55, 61, 64
There are 35 students in a hostel. If the number of students increases by 7, the expenses of the hostel increases by Rs. 42 per day while the average
expenditure per head diminishes by Re. 1. The original expenditure of the hostel is:If \(P=\dfrac{\sum^n_{i=1}(x_i-A)^2}{\sum^n_{i=1}(x_i-\bar x)^2};A\ne\bar x,\) (x̅ = Arithmetic Mean) then P is:
Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
What is the mean age of this group of people?
If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
The average of 25 numbers is 36. If the average of first 13 numbers is 32 and that of the last 13 numbers is 39, then the 13th number is
