Using empirical formula, calculate the **mode** for the following data:

17, 20, 21, 18, 25, 28, 24, 22, 16, 24, 25, 24

Question

Using empirical formula, calculate the **mode** for the following data:

17, 20, 21, 18, 25, 28, 24, 22, 16, 24, 25, 24

Accepted Answer

**Solution:**

1. First, list the data:
   $$17, 20, 21, 18, 25, 28, 24, 22, 16, 24, 25, 24$$

2. Find the frequency of each value:
   - 16: 1
   - 17: 1
   - 18: 1
   - 20: 1
   - 21: 1
   - 22: 1
   - 24: 3
   - 25: 2
   - 28: 1

3. The value with the highest frequency is **24** (appears 3 times).

4. Since the data is raw and not grouped, the mode is the value with the highest frequency, which is **24**.

5. However, the question asks to use the empirical formula for mode:

The empirical formula for mode is:
   $$	ext{Mode} = 3 	imes 	ext{Median} - 2 	imes 	ext{Mean}$$

6. Calculate the mean:
   $$	ext{Mean} = \frac{17 + 20 + 21 + 18 + 25 + 28 + 24 + 22 + 16 + 24 + 25 + 24}{12}$$
   $$= \frac{264}{12} = 22$$

7. Calculate the median:
   - Arrange data in ascending order:
     $$16, 17, 18, 20, 21, 22, 24, 24, 24, 25, 25, 28$$
   - Number of observations, $n=12$ (even)
   - Median = average of 6th and 7th terms
   - 6th term = 22, 7th term = 24
   $$	ext{Median} = \frac{22 + 24}{2} = 23$$

8. Calculate mode using empirical formula:
   $$	ext{Mode} = 3 	imes 23 - 2 	imes 22 = 69 - 44 = 25$$

**Answer:** $25$

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