In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is
120
This problem asks us to find the mode of an asymmetrical distribution given its mean and median. In symmetrical distributions, the mean, median, and mode are equal. However, in asymmetrical (skewed) distributions, they typically differ, and there is an empirical relationship that often holds approximately true, especially for moderately skewed distributions.
The empirical formula relating the mean, median, and mode in a skewed distribution is:
\[ \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \]
We are given the following values:
- Mean = 270
- Median = 220
Now, we can substitute these values into the empirical formula to estimate the mode:
\[ \text{Mode} \approx 3 \times 220 - 2 \times 270 \]
First, calculate the products:
- \(3 \times 220 = 660\)
- \(2 \times 270 = 540\)
Next, substitute these results back into the formula:
\[ \text{Mode} \approx 660 - 540 \]
Perform the subtraction:
\[ \text{Mode} \approx 120 \]
Based on the empirical relationship for asymmetrical distributions, the estimated mode of the data is 120.
| Measure of Central Tendency | Value |
|---|---|
| Mean | 270 |
| Median | 220 |
| Mode (Calculated) | 120 |
Let's compare our calculated mode with the given options:
- 120
- 220
- 280
- 370
The calculated value of 120 matches the first option.
Understanding Skewed Distributions and Central Tendencies
In a perfectly symmetrical distribution, the mean, median, and mode coincide at the peak of the distribution. However, when a distribution is asymmetrical or skewed, these measures of central tendency are pulled in different directions by the tails of the distribution.
- Positively Skewed Distribution: The tail is longer on the right side. Typically, Mean > Median > Mode.
- Negatively Skewed Distribution: The tail is longer on the left side. Typically, Mode > Median > Mean.
In this problem, the Mean (270) is greater than the Median (220). This pattern (Mean > Median) is characteristic of a positively skewed distribution. Following the typical pattern for positive skew (Mean > Median > Mode), the mode should be the smallest of the three values, which aligns with our calculated mode of 120 (since 270 > 220 > 120).
Revision Table: Key Concepts in Statistics
| Term | Definition | Use Case |
|---|---|---|
| Mean | The arithmetic average. Sum of all values divided by the number of values. | Best for symmetrical distributions, sensitive to outliers. |
| Median | The middle value when the data is ordered. | Best for skewed distributions or data with outliers, represents the 50th percentile. |
| Mode | The value that appears most frequently. | Useful for categorical data or finding the most typical value, can have multiple modes or none. |
| Asymmetrical Distribution (Skewed) | A distribution where data is not evenly distributed around the mean, median, or mode. | Requires careful consideration of which measure of central tendency is most appropriate. |
Additional Information on Central Tendency Relationships
The empirical formula \( \text{Mode} \approx 3 \times \text{Median} - 2 \times \text{Mean} \) is a useful rule of thumb but it is not a mathematically derived identity. It works well for distributions that are unimodal (have a single peak) and moderately skewed. For distributions that are heavily skewed, multimodal, or have specific mathematical forms (like exponential or Pareto distributions), this relationship might not hold precisely.
Understanding the relative positions of the mean, median, and mode is crucial for interpreting the shape of a distribution. The median is generally considered a robust measure of central tendency as it is less affected by extreme values (outliers) compared to the mean. The mode tells us about the most frequent observation, which can be very informative for certain types of data.
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