Points to Remember:
- Arithmetic mean calculation involves summing the products of midpoints and frequencies.
- The missing frequency can be found by setting up an equation using the given mean.
- Accurate midpoint calculation is crucial for correct mean calculation.
Introduction:
This question requires a factual and analytical approach to determine a missing frequency in a frequency distribution given the arithmetic mean. Frequency distributions are used to summarize data by grouping values into classes and counting their occurrences (frequencies). The arithmetic mean, or average, is a measure of central tendency calculated by summing all values and dividing by the number of values. In this case, we are given the arithmetic mean (30) and most of the frequencies, allowing us to solve for the missing frequency.
Body:
1. Calculating Midpoints:
First, we need to determine the midpoints of each class interval. The midpoint is calculated as (Upper limit + Lower limit) / 2.
| Class Interval | Frequency (f) | Midpoint (x) | fx |
|—|—|—|—|
| 0-10 | 5 | 5 | 25 |
| 10-20 | 6 | 15 | 90 |
| 20-30 | 10 | 25 | 250 |
| 30-40 | ? (Let’s call it ‘f’) | 35 | 35f |
| 40-50 | 13 | 45 | 585 |
2. Setting up the Equation:
The formula for the arithmetic mean (xÌ) is:
xÌ = Σ(fx) / Σ(f)
We are given that xÌ = 30. The sum of frequencies (Σf) is 5 + 6 + 10 + f + 13 = 34 + f. The sum of the products of frequencies and midpoints (Σfx) is 25 + 90 + 250 + 35f + 585 = 950 + 35f.
Therefore, we can set up the equation:
30 = (950 + 35f) / (34 + f)
3. Solving for the Missing Frequency (f):
Now we solve for ‘f’:
30(34 + f) = 950 + 35f
1020 + 30f = 950 + 35f
1020 – 950 = 35f – 30f
70 = 5f
f = 14
4. Verifying the Solution:
Let’s verify our solution by recalculating the mean with f = 14:
Σf = 34 + 14 = 48
Σfx = 950 + 35(14) = 1440
xÌ = 1440 / 48 = 30
The calculated mean matches the given mean, confirming that the missing frequency is 14.
Conclusion:
The missing frequency in the given frequency distribution is 14. This was determined by utilizing the formula for the arithmetic mean, setting up an equation based on the provided mean and known frequencies, and solving for the unknown frequency. The solution was then verified by recalculating the mean using the found frequency. This analytical approach highlights the importance of understanding statistical concepts like frequency distributions and measures of central tendency for data analysis. Further analysis could involve exploring the distribution’s shape and other descriptive statistics to gain a more comprehensive understanding of the data. Accurate data analysis is crucial for informed decision-making in various fields.
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