If Mean > Median > Mode, the distribution is:
right-skewed
Understanding Statistical Distributions and Skewness
In statistics, the shape of a distribution describes how the data is spread out. One important aspect of the shape is its symmetry or asymmetry, which is known as skewness. Skewness tells us whether the data is concentrated more on one side than the other.
The relationship between the three common measures of central tendency — Mean, Median, and Mode — is a good indicator of the skewness of a distribution.
Measures of Central Tendency
- Mean: The average of all the data points. It is affected by extreme values.
- Median: The middle value in a dataset when it is ordered from least to greatest. It is less affected by extreme values than the mean.
- Mode: The value that appears most frequently in the dataset.
Analysing the Given Condition: Mean > Median > Mode
The question provides a specific relationship between the measures of central tendency: Mean is greater than Median, which is greater than Mode ($\text{Mean} > \text{Median} > \text{Mode}$). Let's consider how this relationship applies to different types of distributions:
Types of Skewness
- Normal Distribution: This is a perfectly symmetrical distribution where the Mean, Median, and Mode are all equal ($\text{Mean} = \text{Median} = \text{Mode}$). The data is evenly distributed around the center.
- Left-Skewed Distribution (Negative Skew): In a left-skewed distribution, the tail is on the left side. The majority of the data is concentrated on the right. The relationship between the measures of central tendency is typically $\text{Mode} > \text{Median} > \text{Mean}$. The mean is pulled down by the lower values in the left tail.
- Right-Skewed Distribution (Positive Skew): In a right-skewed distribution, the tail is on the right side. The majority of the data is concentrated on the left. The relationship between the measures of central tendency is typically $\text{Mean} > \text{Median} > \text{Mode}$. The mean is pulled up by the higher values in the right tail.
Relating the Condition to Distribution Type
The given condition is $\text{Mean} > \text{Median} > \text{Mode}$. Comparing this to the characteristics of different distributions:
- Normal Distribution: $\text{Mean} = \text{Median} = \text{Mode}$ (Does not match the condition)
- Left-Skewed Distribution: $\text{Mode} > \text{Median} > \text{Mean}$ (Does not match the condition)
- Right-Skewed Distribution: $\text{Mean} > \text{Median} > \text{Mode}$ (Matches the condition)
Therefore, when the Mean is greater than the Median, and the Median is greater than the Mode ($\text{Mean} > \text{Median} > \text{Mode}$), the distribution is right-skewed.
| Distribution Shape | Relationship of Mean, Median, Mode | Tail Direction |
|---|---|---|
| Normal (Symmetrical) | $\text{Mean} = \text{Median} = \text{Mode}$ | No tail (symmetrical) |
| Left-Skewed (Negative Skew) | $\text{Mode} > \text{Median} > \text{Mean}$ | Left |
| Right-Skewed (Positive Skew) | $\text{Mean} > \text{Median} > \text{Mode}$ | Right |
Based on the analysis, the distribution where Mean > Median > Mode is a right-skewed distribution.
Revision Table: Key Concepts Review
| Term | Definition/Characteristic | Relevance to Skewness |
|---|---|---|
| Skewness | Measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. | Indicates whether the data tails off to one side. |
| Mean | Arithmetic average. | Sensitive to extreme values, pulled towards the tail. |
| Median | Middle value when ordered. | Less sensitive to extreme values than the mean. |
| Mode | Most frequent value. | Represents the peak of the distribution. |
| Right-Skewed | Tail on the right. | $\text{Mean} > \text{Median} > \text{Mode}$ (usually) |
| Left-Skewed | Tail on the left. | $\text{Mode} > \text{Median} > \text{Mean}$ (usually) |
Additional Information: Skewness Calculation and Interpretation
While the relationship between Mean, Median, and Mode is a good indicator of skewness, especially for unimodal distributions, skewness can also be calculated numerically. Common methods include Pearson's coefficient of skewness (based on Mean and Median or Mean and Mode) or the third standardized moment.
- Pearson's First Coefficient: $\text{Skewness} \approx \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}}$
- Pearson's Second Coefficient: $\text{Skewness} \approx \frac{3 \times (\text{Mean} - \text{Median})}{\text{Standard Deviation}}$
A positive skewness value indicates a right-skewed distribution, a negative value indicates a left-skewed distribution, and a value near zero indicates a symmetrical distribution. The rule $\text{Mean} > \text{Median} > \text{Mode}$ for right-skewness and $\text{Mode} > \text{Median} > \text{Mean}$ for left-skewness are strong general tendencies for unimodal distributions, but exceptions can occur in multi-modal or unusual distribution shapes.
Important Questions from Basics of Probability
If the data are skewed, which option of central tendency measure is the most unreliable indicator?
In a negatively skewed distribution
If the distribution is negatively skewed, then the:
The first four moments about the mean of distribution are 0, μ 2, 0.7 and 18.75. If the distribution is mesokurtic, the value of μ 2, is
The following measures were computed for a moderately symmetrical frequency distribution: mean = 50, coefficient of variation = 35% and Karl Pearson's Coefficient of Skewness = - 0.25. The value of the median of the distribution is:
