Consider the following 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.

a) What is the mean of the data? What is the median?

b) What is the mode of data?

c) What is the midrange of the data?

d) Can you find Q1,Q3?

e) Show a boxplot of the data.

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%.

$$ Upper\ Quartile\ (Q3) = \Biggl[(n + 1) \times \frac 34 \Biggr]^{th} number$$ $$ Upper\ Quartile\ (Q3) = (27 + 1) \times \frac34 = \frac {84}{4} = 21^{th}\ positioned\ number $$ $$ Q3 = 35 $$

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

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

e] Boxplot of Data Points

• Box plots represent the graphical image of the concentration of the data points.

• The box plot is created based on the 5 values as follows:

  • The Minimum Value = 13

  • The First Quartile (Q1) = 20

  • The Median (Q2) = 25

  • The Third Quartile (Q3) = 35

  • The Maximum Value = 70

• The box plot can be drawn either by vertically or horizontally.
• For the given data points Horizontal Box Plot can be drawn as follows: