Question

Consider the following data:

62 61 83 92 67 89 83 50 80 69 95 50 73 71 73 88 66 89 73 51 58 86 73 57 67 80 59 86 62 95 63 88 59 64 54 78 84 77 71 82 82 83 81 97 60 85 61 55 80 72

Make frequency distribution using 5 classes and find the following:

Arithmetic Mean

Median

Mode

All quartiles

3rd Deciles

46th Percentile

Coefficient of Skewness

Semi Inter Quartile Range (SIQR)

Standard Deviation

Mean Deviation

Variance

Sketch Histogram

Sketch OGIVE

Answer 1

Arithmetic Mean:

$\mu=\frac{\sum x_i}{n}=\frac{3664}{50}=73.28$

Put numbers in increasing order:

$50,50,51,54,55,57,58,59,59,60,61,61,62,62,63,64,66,67,67,69,71,71,72,$

$73,73,73,73,77,78,80,80,80,81,82,82,83,83,83,84,85,86,86,88,88,89,89,$

$92,95,95,97$

Median (the number in the middle of the list):

$m=73$

Mode (the number which appears most often in a set of numbers):

$M=73$

Quartiles:

1st quartile is number on the (N/4)th place:

$Q_1=61$

2nd quartile is median:

$Q_2=m=73$

3rd quartile is number on the (3N/4)th place:

$Q_3=84$

3rd Decile:

$D_3=value\ of\ 3(50+1)/10=1.7th\ number$, which is 0.7 of the way between 1st and 2nd numbers (50 and 50).

$D_3=50$

46th Percentile:

The 46th percentile is the lowest number in a data set that is greater than 46% of the numbers:

index=$0.46\cdot50=23$

$Q_{46}=N_{24}=73$

Coefficient of Skewness:

$S_k=3\frac{\mu-m}{\sigma}$

$\sigma=\sqrt{\frac{\sum (x_i-\mu)^2}{n}}=\sqrt{\frac{7971.9216}{50}}=12.63$

$S_k=\frac{72.28-73}{12.63}=0.022$

Semi Inter Quartile Range (SIQR):

$SIQR=\frac{Q_3-Q_1}{2}=\frac{84-61}{2}=11.5$

Standard Deviation:

$\sigma=12.63$

Mean Deviation:

$MD=\frac{\sum |x_i-\mu|}{n}=\frac{555.12}{50}=11.10$

Variance:

$V=\sigma_2=12.63^2=159.52$

Histogram:

OGIVE: