Question

Consider the following ungrouped data:

41 46 7 46 32 5 14 28 48 49 8 49 48 25 41 8 22 46 40 48

Find the following:

a)  Arithmetic mean

b)  Geometric mean

c)  Harmonic mean

d)  Median

e)  Mode

f)  Range

g)  Mean deviation

h)  Variance

i)  Standard Deviation

Answer 1

$$5, 7, 8, 8, 14, 22, 25, 28, 32, 40,$$

$$41, 41, 46, 46, 46, 48, 48, 48, 49,49$$

a)

Arithmetic mean$=\dfrac{\displaystyle\sum_{i=1}^{20}x_i}{20}=\dfrac{651}{20}=32.55$

b)

Geometric mean$=\sqrt[20]{\displaystyle\prod_{i=1}^{20}x_i}\approx26.3908$

c)

Harmonic mean$=\dfrac{20}{\displaystyle\sum_{i=1}^{20}\dfrac{1}{x_i}}\approx18.8854$

d)

Median = $\dfrac{40+41}{2}=40.5$

e)

Modes are 46 and 48. Each appeared 3 times

f) Range $=49-5=44$

g)  Mean deviation

Mean absolute deviation $MAD$

$$MAD=\dfrac{1}{20}\displaystyle\sum_{i=1}^{20}|x_i-\bar{x}|=14.395$$

h)  Variance

The variance of a sample is:

$$s^2=\dfrac{1}{20-1}\displaystyle\sum_{i=1}^{20}(x_i-\bar{x})^2\approx 270.99736842$$

i)  Standard Deviation

$$s=\sqrt{s^2}\approx\sqrt{270.99736842}\approx16.4620$$