The mean and median of the distribution is 12 and 15. Then the mode equals to:
21
Finding the Mode from Mean and Median
The question asks us to find the mode of a distribution given its mean and median. In statistics, for moderately skewed distributions, there is an empirical relationship between the mean, median, and mode. This relationship is often expressed by the formula:
Mode \(\approx\) 3 \(\times\) Median - 2 \(\times\) Mean
This formula is particularly useful when the distribution is not symmetrical but also not extremely skewed, as it provides a good estimate for the mode.
Applying the Empirical Formula
We are given the following values:
- Mean = 12
- Median = 15
Now, we can substitute these values into the empirical formula to calculate the estimated mode:
Mode \(\approx\) 3 \(\times\) Median - 2 \(\times\) Mean
Mode \(\approx\) 3 \(\times\) 15 - 2 \(\times\) 12
Mode \(\approx\) 45 - 24
Mode \(\approx\) 21
Based on the empirical formula, the estimated mode of the distribution is 21.
Comparing with Options
Let's compare our calculated mode with the given options:
- 21
- 24
- 15
- 18
Our calculated value of 21 matches Option 1.
Summary of Calculation
We used the empirical relationship between the measures of central tendency to estimate the mode.
Given:
- Mean = 12
- Median = 15
Formula: Mode \(\approx\) 3 * Median - 2 * Mean
Calculation:
Mode \(\approx\) 3 * 15 - 2 * 12
Mode \(\approx\) 45 - 24
Mode \(\approx\) 21
Therefore, the mode is approximately 21.
Revision Table: Mean, Median, and Mode Relationship
| Measure | Given Value | Role in Formula |
|---|---|---|
| Mean | 12 | Used in the formula: 2 \(\times\) Mean |
| Median | 15 | Used in the formula: 3 \(\times\) Median |
| Mode | To be found | Estimated using the formula |
Additional Information: Measures of Central Tendency
Mean, median, and mode are three common measures of central tendency that represent a typical value in a dataset. Understanding their relationship is important in statistics.
- Mean: The average of all values. It is sensitive to extreme values (outliers).
- Median: The middle value in a dataset that has been ordered from least to greatest. It is not affected by extreme values.
- Mode: The value that appears most frequently in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.
The empirical formula Mode \(\approx\) 3 * Median - 2 * Mean is based on observed data and is a rule of thumb, not a strict mathematical identity that holds for all distributions. It works best for distributions that are moderately skewed. In a perfectly symmetrical distribution, the mean, median, and mode are all equal.
- For a positively skewed distribution (tail on the right), typically Mean > Median > Mode.
- For a negatively skewed distribution (tail on the left), typically Mode > Median > Mean.
The formula used helps estimate the mode's position relative to the mean and median in such skewed distributions.
Important Questions from Measures of Central Tendency
The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is
The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at
The observations 4, 1, 4, 3, 6, 2, 1, 3, 4, 5, 1, 6 are outputs of 12 dices thrown simultaneously. If m and M are means of lowest 8 observations and highest 4 observations respectively, then what is (2m + M) equal to ?
The mean of the series x 1, x 2… x nis X̅. If x 2 is replaced by λ, then what is the new mean?
The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?
