*The given Data Points* **= 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70**

*Total Data Points (n)* **= 27**

**a] Mean and Median of the Data Points**

**Mean -**

- Mean is the arithmetic average of data points.
- This is the addition of the numbers of data points and dividing by the total data points.

$$ Mean\ of\ Data\ Points = \frac{Addition\ of\ all\ the\ Data\ Points}{Total\ Data\ Points}$$

$$ Mean of Data Points = \frac {\{13 + 15 + 16 + 16 + 19 + 20 + 20 + 21 + 22 + 22 + 25 + 25 + 25 + 25 + 30 + 33 + 33 + 35 + 35 + 35 + 35 + 36 + 40 + 45 + 46 + 52 + 70\}}{27} $$

$$ Mean\ of\ Data\ Points = \frac{809}{27} = 29.96 $$

**Median -**

- The median is the middle number in the data points when the numbers are listed in either ascending or descending order.
- If data points are not listed in any order then first arrange them in ascending order and then find out the Median.

*Median when total Data Points (n) are ODD:*

$$ Median = \Biggl(\frac{n + 1}{2}\Biggr)^{th}\ number $$

*Median when total Data Points (n) are EVEN:*

$$ Median = \frac{\bigl(\frac n2\bigr)^{th} number + \bigl(\frac {n + 1}{2}\bigr)^{th} number}{2} $$

- Here, the total data points are
**27 which means ODD.**

$$ Median\ of\ Data\ Points = \Biggl(\frac{27 + 1}{2}\Biggr)^{th}\ number = 14^{th}\ positioned\ number$$

*Therefore,* $$ Median = 25 $$

**b] Mode of Data Points**

- The mode is the most frequently occurring number in the data points.
- Here, in the given data points numbers
**25 and 35** are modes with the *most frequent occurrence count of 4.*

$$ Mode\ of\ Data\ Points = 25\ and\ 35\ (Occurence\ Count\ 4) $$

**c] Midrange of Data Points**

- Midrange is the difference between the highest and lowest values in the data points.
- It shows the halfway between the minimum and maximum numbers of the data points.

$$ Midrange\ of\ Data\ Points = \frac{Maximum\ Number\ in\ Data\ Points + Minimum\ Number\ in\ Data\ Points}{2} $$

$$ Midrange\ of\ Data\ Points = \frac{70 + 13}{2} = \frac{83}{2}$$

$$ Midrange\ of\ Data\ Points = 41.5$$

**d] Q1, Q3 of Data Points**

- Q1 and Q3 represent the Quartiles.
- In statistical measure, a quartile, is one type of quantile of three points (Q1, Q2, & Q3) that divides sorted data points into four equal groups in terms of count of numbers, each representing a fourth of the distributed sampled population.
- There are three quartiles as follows:

**The First Quartile (Q1)** - *It is a 1*^{st} quartile or lower quartile that separates the lowest 25% of data from the highest 75%.

$$ Lower\ Quartile\ (Q1) = \Biggl[(n + 1) \times \frac 14 \Biggr]^{th} number$$
$$ Q1 = (27 + 1) \times \frac 14 = \frac {28}{4} = 7^{th}\ positioned\ number $$
$$ Q1 = 20 $$

**The Second Quartile (Q2)** - *It is a 2*^{nd} quartile or middle quartile also same as **Median** it divides numbers into 2 equal parts.

$$ Middle\ Quartile\ (Q2) = \Biggl[(n + 1) \times \frac 24 \Biggr]^{th} number $$
$$ Q2 = (27 + 1) \times \frac 24 = \frac {56}{4} = 14^{th}\ positioned\ number $$
$$ Q2 =25 $$

**The Third Quartile (Q3)** - *It is a 3*^{rd} quartile or the upper quartile that separate the highest 25% of data from the lowest 75%.

$$ Upper\ Quartile\ (Q3) = \Biggl[(n + 1) \times \frac 34 \Biggr]^{th} number$$
$$ Upper\ Quartile\ (Q3) = (27 + 1) \times \frac {84}{4} = 21^{th}\ positioned\ number $$
$$ Q3 = 35 $$

- Based on Q1 & Q3 values
*Interquartile Range* also calculated as follows:

$$ Interquartile\ Range = Q3 – Q1 = 35 - 20 = 15 $$

**e] Boxplot of Data Points**