Question

Define/Explain the following concepts.

1. Arithmetic mean
2. Median.
3. Mode
4. Quartiles
5. Deciles.
6. Percentiles
7. Range

Answer 1

1. Arithmetic mean - The arithmetic mean refers to the value calculated by dividing the sum of the values of the given items by the total number of items present in a series. Mathematically, it is measured as:

$\overline{X}=\frac{x_1+x_2+x_3+......+x_n}{N}$

Here, $\overline{X}$ is the arithmetic mean.

2. Median - The median is the middlemost value of a set of data when arranged in an array (ascending or descending). The median divides the data into two equal groups. Mathematically, for an individual series, it is measured as:

M = Size of the $\left(\frac{N+1}{2}\right)$ ^th item
Here,
M = Median
N = Total number of items

3. Mode - The mode is the observation or value that repeats itself the maximum number of times or that has the maximum frequency. It is easily measured by merely looking and locating the value with the highest frequency.

4. Quartiles - Quartiles are the values of the given set of data that divide the whole data or the set of observations into four equal segments.
There are three quartiles in data known as the first, second and third quartiles.
Mathematically,

$$\begin{gathered} Q_1={\left(\frac{N+1}{4}\right)}^{th}\mathrm{observation} \\ {Q}_2=2{\left(\frac{\mathrm{N}+1}{4}\right)}^{th}\mathrm{observation} \\ {Q}_3=3{\left(\frac{\mathrm{N}+1}{4}\right)}^{th}\mathrm{observation} \end{gathered}$$

Here, N is the total number of items.

5. Deciles - Deciles are the values of the given set of data that divide the whole data into 10 equal groups. There are 9 deciles in a set of data that divide the set of observations into 10 parts of the same size, namely $D_1{D}_2{D}_3{D}_4{D}_5{D}_6{D}_7{D}_8{D}_9$.
Mathematically,
$D_1={\left(\frac{N+1}{10}\right)}^{th}item$

$D_2=2{\left(\frac{N+1}{10}\right)}^{th}item$

$D_3=3{\left(\frac{N+1}{10}\right)}^{th}item$

$D_4=4{\left(\frac{N+1}{10}\right)}^{th}item$

$D_5=5{\left(\frac{N+1}{10}\right)}^{th}item$

$D_6=6{\left(\frac{N+1}{10}\right)}^{th}item$

$D_7=7{\left(\frac{N+1}{10}\right)}^{th}item$

$D_8=8{\left(\frac{N+1}{10}\right)}^{th}item$

$D_9=9{\left(\frac{N+1}{10}\right)}^{th}item$

Here, N is the total number of items.

6. Percentile - A percentile divides the given set of data into 100 equal parts. There are a total of 99 percentiles that help in dividing the data into 100 parts by giving 99 dividing points, namely $P_1{P}_2{P}_3.............P_{99}$.
Mathematically,
$P_1={\left(\frac{N+1}{100}\right)}^{th}item$

$P_2=2{\left(\frac{N+1}{100}\right)}^{th}item$

$P_3=3{\left(\frac{N+1}{100}\right)}^{th}item$
.
.
.
.
$P_{99}=99{\left(\frac{N+1}{100}\right)}^{th}item$
Here, N is the total number of items.

7. Range - The range is defined as the span of distribution. It is calculated by finding the difference between the smallest and largest values of the given set of distribution.
Range = Largest value – Smallest value
Or,
R = L – S