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15: Find the sample variance and standard deviation for frequency distribution of height in cms of students in an AU given below. Heights in cms 15 15 0 2 4 15 156 15 16 162 16 16 8 0 4 6 Number of students 28 40 52 100 60 48 32 20 7​

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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

----------------------------------------------

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

------------------------------------------------------------------------------------------------

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:

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