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Juliette [100K]
3 years ago
11

In a data distribution, the first quartile, the median and the means are 30.8, 48.5 and 42.0

Mathematics
1 answer:
kolbaska11 [484]3 years ago
5 0

Answer:

Q_3 = 56.45 --- The third quartile

Var = 370.18 -- Variance

Step-by-step explanation:

Given

Q_1  =  30.8 -- First quartile

Q_2 =  48.5 --- Median

\bar x = 42 --- Mean

Skp = -0.38 --- Coefficient of skewness

Solving (9): The third quartile Q_3

This is calculated from

Skp =  \frac{Q_1   + Q_3  - 2Q_2}{Q_3 - Q_1}

So, we have:

-0.38 =  \frac{30.8 + Q_3- 2*48.5}{Q_3 - 30.8}

Cross Multiply

-0.38 (Q_3 - 30.8)=  30.8 + Q_3- 2*48.5

Open bracket

-0.38Q_3 + 11.704=  30.8 + Q_3- 97.0

Collect like terms

-0.38Q_3 -Q_3=  30.8 - 97.0- 11.704

-1.38Q_3=  -77.904

Divide both sides -1.38

Q_3 = \frac{-77.904}{-1.38}

Q_3 = 56.45 --- approximated

Solving (b): The variance

First, calculate the standard deviation from:

3IQR  = 4SD

IQR= Q_3 - Q_1

So:

3IQR  = 4SD

3(Q_3 - Q_1) = 4SD

Make SD the subject

SD = \frac{3}{4}(Q_3 - Q_1)

SD = \frac{3}{4}(56.45 - 30.8)

SD = \frac{3}{4}*25.65

SD = \frac{3*25.65}{4}

SD = \frac{76.95}{4}

SD = 19.24

So, the variance is:

Var = SD^2

Var = 19.24^2

Var = 370.18

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