Answer:
To find the sample variance and standard deviation for the given frequency distribution, we need to calculate the mean height first. We can do this by summing up the product of each height value and its corresponding frequency, and then dividing by the total number of students.
Height (cms) | Frequency | Frequency * Height
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0 | 28 | 0
2 | 40 | 80
4 | 52 | 208
6 | 100 | 600
8 | 60 | 480
15 | 48 | 720
16 | 32 | 512
156 | 20 | 3120
162 | 7 | 1134
Total number of students: 347
Sum of Frequency * Height: 7444
Mean height = (Sum of Frequency * Height) / Total number of students
= 7444 / 347
≈ 21.45
Now, let's calculate the sample variance using the formula:
Sample variance = (Sum of (Frequency * (Height - Mean Height)^2)) / (Total number of students - 1)
Height (cms) | Frequency | Height - Mean | (Height - Mean)^2 | Frequency * (Height - Mean)^2
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0 | 28 | -21.45 | 459.8025 | 12874.47
2 | 40 | -19.45 | 378.8025 | 15152.10
4 | 52 | -17.45 | 304.8025 | 15834.13
6 | 100 | -15.45 | 238.8025 | 23880.25
8 | 60 | -13.45 | 180.8025 | 10848.15
15 | 48 | -6.45 | 41.8025 | 2003.92
16 | 32 | -5.45 | 29.7025 | 950.48
156 | 20 | 134.55 | 18094.5025 | 361890.05
162 | 7 | 140.55 | 19748.5025 | 138239.52
Sum of Frequency * (Height - Mean)^2: 593672.77
Sample variance = (Sum of Frequency * (Height - Mean)^2) / (Total number of students - 1)
= 593672.77 / (347 - 1)
≈ 1714.94
Finally, we can calculate the sample standard deviation by taking the square root of the sample variance:
Sample standard deviation = √(Sample variance)
= √(1714.94)
≈ 41.40
Therefore, the sample variance for the given frequency distribution is approximately 1714.94 and the sample standard deviation is approximately 41.40.
Explanation: