If a histogram and a frequency polygon are drawn for the frequency distribution:
| Class | 1-5 | 6-10 | 11-15 | 16-20 | 21-25 |
| :-------- | :-- | :--- | :---- | :---- | :---- |
| Frequency | 2 | 4 | 6 | 5 | 3 |
Answer (2)
Calculate class midpoints: 3 , 8 , 13 , 18 , 23 .
Calculate the mean: 20 275 = 13.75 .
Calculate the approximate median: 14.33 .
Calculate the approximate mode: 13 .
Mean = 13.75 , Median = 14.33 , Mode = 13
Explanation
Analyze the problem and data We are given a frequency distribution table and asked to analyze it. The table provides class intervals and their corresponding frequencies. Our goal is to understand the distribution by calculating key statistical measures and visualizing the data.
Calculate class midpoints First, we need to find the midpoint of each class interval. The midpoint is calculated as (Lower Limit + Upper Limit) / 2.
List class intervals and midpoints The class intervals and their midpoints are:
1-5: 2 1 + 5 = 3
6-10: 2 6 + 10 = 8
11-15: 2 11 + 15 = 13
16-20: 2 16 + 20 = 18
21-25: 2 21 + 25 = 23
Calculate the mean Next, we calculate the mean of the frequency distribution. The mean is given by: Mean = ∑ Frequency ∑ ( Midpoint × Frequency )
Compute the mean Using the midpoints and frequencies, we calculate the mean: Mean = 2 + 4 + 6 + 5 + 3 ( 3 × 2 ) + ( 8 × 4 ) + ( 13 × 6 ) + ( 18 × 5 ) + ( 23 × 3 ) = 20 6 + 32 + 78 + 90 + 69 = 20 275 = 13.75
Calculate the variance Now, let's calculate the variance. The variance is given by: Variance = ∑ Frequency ∑ (( Midpoint − Mean ) 2 × Frequency )
Compute the variance Using the midpoints, frequencies, and the calculated mean, we find the variance:
Variance = 20 (( 3 − 13.75 ) 2 × 2 ) + (( 8 − 13.75 ) 2 × 4 ) + (( 13 − 13.75 ) 2 × 6 ) + (( 18 − 13.75 ) 2 × 5 ) + (( 23 − 13.75 ) 2 × 3 )
Variance = 20 ( 115.5625 × 2 ) + ( 33.0625 × 4 ) + ( 0.5625 × 6 ) + ( 18.0625 × 5 ) + ( 85.5625 × 3 )
Variance = 20 231.125 + 132.25 + 3.375 + 90.3125 + 256.6875 = 20 713.75 = 35.6875
Calculate the standard deviation The standard deviation is the square root of the variance: Standard Deviation = Variance = 35.6875 ≈ 5.97
Approximate the median The median is the middle value of the dataset. Since we have grouped data, we need to approximate the median using the formula: Median = L + f 2 N − CF × h Where:
L is the lower boundary of the median class
N is the total frequency
CF is the cumulative frequency before the median class
f is the frequency of the median class
h is the class width
Compute the median The total frequency N = 20, so N/2 = 10. The cumulative frequencies are 2, 6, 12, 17, 20. The median class is 11-15 because its cumulative frequency (12) is the first to exceed 10.
L = 11, CF = 6, f = 6, h = 5
Median = 11 + 6 10 − 6 × 5 = 11 + 6 4 × 5 = 11 + 6 20 = 11 + 3.33 = 14.33
Find the mode The mode is the value that appears most frequently. In grouped data, we consider the modal class, which is the class with the highest frequency. The modal class is 11-15 with a frequency of 6. We approximate the mode as the midpoint of this class.
Mode = 2 11 + 15 = 13
State the results The mean is 13.75, the variance is 35.6875, the standard deviation is approximately 5.97, the median is approximately 14.33, and the mode is 13.
Examples
Understanding frequency distributions is crucial in many real-world scenarios. For instance, in market research, analyzing the distribution of customer ages can help tailor marketing strategies. Similarly, in environmental science, frequency distributions of pollution levels can aid in identifying sources and implementing control measures. In education, analyzing the distribution of test scores helps teachers understand the effectiveness of their teaching methods and adjust them accordingly. These statistical measures provide valuable insights for informed decision-making.
In this analysis of the frequency distribution, the mean is calculated as 13.75, the median is approximately 14.33, and the mode is 13. The calculations involve finding midpoints for each class and using statistical formulas for these measures. Understanding these concepts is crucial for interpreting data effectively.
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