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Consider the following frequency distribution.

Class Frequency
2 up to 4 32
4 up to 6 48
6 up to 8 92
8 up to 10 32
a.
Calculate the population mean. (Round your answer to 2 decimal places.)

Population mean
b.
Calculate the population variance and the population standard deviation. (Round your intermediate calculations to 4 decimal places and final answers to 2 decimal places.)

Population variance
Population standard deviation

User Dallion
by
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1 Answer

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2 votes

Answer: Restating the Frequency distribution clearly

Class Frequency

2 up to 4 32

4 up to 6 48

6 up to 8 92

8 up to 10 32

a. Population mean (μ) = 6.22

b. Population Variance = 3.88

c. Standard deviation = 1.97

Explanation:

1. To solve this, we first of all have to form a frequency table, and in order to get the different components of the table, we have to state the formula of the most complex part of the question that is the standard deviation

Standard deviation (S) =√ [∑f(m-μ)² / ∑f - 1].

2. next, we will form the frequency table as shown below making sure to include all the components of the standard deviation formula.

Class f m f×m μ (m-μ) (m-μ)² f(m-μ)²

2-4 32 3 96 6.22 -3.22 10.3684 331.7888

4-6 48 5 240 6.22 -1.22 1.4884 71.4432

6-8 92 7 644 6.22 1.22 1.4884 136.9328

8-10 32 9 288 6.22 2.78 7.7284 247.3088

204 1268 787.4736

where:

f = frequency

m = midpoint

μ = mean

3. After getting f, and m, we can calculate the mean which is also known as the average value, as shown below:

mean (μ) : ∑(f × m) / ∑f Note that the symbol "∑" stands for "sum of"

∴ μ =
(1268)/(204) = 6.2157

Rounding to 2 decimal places μ = 6.22

4. Next let us define what standard deviation is; the standard deviation is the extent to which a value differ from the mean value. The formula is shown below.

Standard deviation (S) =√ [∑f(m-μ)² / ∑f - 1]

S
\sqrt{(787.4736)/(204 - 1) } = \sqrt{(787.4736)/(203) } \\ \\ = √(3.8792) = 1.9696

∴ S = 1.97 (to 2 decimal places)

5. finally, the variance is the square of the standard deviation and it is shown thus:

Variance (S²) =
(√(3.8792)) ^(2) = 3.8792 = 3.88 (2 decimal places).

User Andyz Smith
by
3.3k points