# Range and Quartiles

** Published:**

This lesson covers Range and Quartiles.

Sources:

- https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch12/5214890-eng.htm

# Range

- Difference between largest and smallest observed values
- Actual spread of the data, including outliers
- Can be expressed as lower limit - upper limit e.g. $1-10$ or as range upper limit - lower limit e.g. $9$
- Doesn’t measure spread of the majority of the data spread since it only measures spread between lowest and highest value

# Quartiles

Median divides the dataset in two equal sets.

Lower Quartile ($Q1$) - first quartile

- middle of first set
- $25\%$ of values are smaller than $Q1$ and $75\%$ values are higher than $Q1$

Upper Quartile ($Q3$) - third quartile

- middle of second set
- $75\%$ of values are smaller than $Q3$ and $25\%$ values are higher than $Q3$

Second Quartile ($Q2$)

- Median of the data set

Example

Data: 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36

Ordered data: 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49

Median: 41

- 1st part: 6, 7, 15, 36, 39
- 2nd part: 41, 43, 43, 47, 49

middle value not included since total odd number else it is included in both parts

Upper quartile: 43

Lower quartile: 15

Second quartile: 41

# Interquartile range

- Difference between upper quartile and lower quartile
- Measures spread of the data
- Spans $50\%$ of the data and eliminate influence of outliers by removing highest and lowest quarters
- $IQR = Q3 - Q1$

# Example

- Question: A year ago, Angela began working at a computer store. Her supervisor asked her to keep a record of the number of sales she made each month. The following data set is a list of her sales for the last $12$ months: $34, 47, 1, 15, 57, 24, 20, 11, 19, 50, 28, 37$. Use Angela’s sales records to find:
- the median
- the range
- the upper and lower quartiles
- the interquartile range

- Answer:
- $Median = 26$
- $Range = 56$
- $Q1 = 17$
- $Q3 = 42$
- $IQR = 25$

# Semi-quartile Range

- One half of the difference between $Q3$ and $Q1$ i.e. $\frac{Q3-Q1}{2}$
- If the Distribution is Symmetric
- $50\%$ of the data is covered by $Median \pm SQR$
- Not true if it skewed

- $SQR$ Is hardly affected by higher values. Good measure of spread for skewed distribution
- Rarely used for datasets that have normal distribution
- Standard Deviation is used

# Box and Whisker Plot

Source: mathsisfun.com |

- Question: $4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11$
- Answer:
- In order: $3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18$
- Cut in Quarters:
- $3, 4, 4, 4, 7, 10,~~~~ 11, 12, 14, 16, 17, 18$
- $Q2=10.5;~ Q1=4;~ Q3=15$
- $3, 4, 4,~~~ 4, 7, 10,~~ 11, 12, 14,~~~ 16, 17, 18$

- $Min=3;~ Max=18$
- $IQR = 11$

## Five -Number Summary

- $Q2=10.5;~ Q1=4;~ Q3=15$
$Min=3;~ Max=18$

## MS Excel:

- Select Data and Insert -> Statistical -> Box and Whisker

More Questions at: https://www.mathsisfun.com/data/quartiles.html