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Wize High School Grade 9 Math Textbook > Data Analysis

Measures of Spread - Quartiles & Box Plots

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Quartiles

Wize Concept
Recall that a variable is an attribute that we measure, such as height, eye colour, age, etc.

When we are collecting numerical data for one variable, we will end up with a list of numbers called a data set. We can use quartiles and box plots to help us represent and visualize the spread of our data set.

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Quartiles

When we have a one-variable numerical data set that is ordered from smallest to largest
  • the first or lower quartile (Q1Q_1) is the spot that separates the first quarter of the data from the rest of the data
  • the second quartile or Median (Q2Q_2) is the spot that separates the first half of the data from the rest of the data
  • the third or upper quartile (Q3Q_3) is the spot that separates the last quarter of the data from the rest of the data

How do we Find the Quartiles?

  1. Order the data from smallest to largest
  2. Find the median Q2Q_2 -- this divides the data into two equal parts
  3. Find the median of the first half of the data -- this is the first quartile Q1Q_1
  4. Find the median of the second half of the data -- this is the third quartile Q3Q_3

Example
Find the quartiles for the dataset 2,  2,  7, 3,  11,  9,  82,~~2,~~7,~3,~~11,~~9,~~8.
  1. Order the data from smallest to largest: 2,  2,  3,  7,  8, 9,  112,~~2,~~3,~~7,~~8,~9,~~11
  2. Find the median: 2,  2,  3,  7⏟median Q2,  8, 9,  112,~~2,~~3,~~\underbrace{7}_{\text{median }Q_2},~~8,~9,~~11
  3. Find the median of the first half of the data: 2,  2⏟first quartile Q1,  32,~~\underbrace{2}_{\text{first quartile }Q_1},~~3
  4. Find the median of the second half of the data: 8,  9⏟second quartile Q3,  118,~~\underbrace{9}_{\text{second quartile }Q_3},~~11
So, the first, second, and third quartiles divide our data into 4 part, each part with an equal number of data points:
2,  2⏟first quartile Q1,  3,  7⏟median Q2,  8, 9⏟third quartile Q3,  11\boxed{2,~~\underbrace{2}_{\colorTwo{\text{first quartile }Q_1}},~~3,~~\underbrace{7}_{\colorTwo{\text{median }Q_2}},~~8,~\underbrace{9}_{\colorTwo{\text{third quartile }Q_3}},~~11}

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Box Plots (Box-and-Whisker Plots)

A box plot is a graph that is used to show how numerical (quantitative) data is distributed (how spread out it is).


Box plots can be horizontal (like in the picture above) or they can be vertical if you choose to organize your data from smallest (lowest point) to largest (highest point)

50% of the dataset will be captured within the big box (meaning that half of your data points will be within the big box)

Extra Definitions

  • The difference between the maximum and minimum data points is called the range -- this tells us the distance the data points cover
  • The different between the Q1 and Q3 is called the interquartile range -- this tells us the distance the middle 50% of data covers
  • Sometimes, a dataset can have outliers, which are data points are very different than the rest of the data points. Outliers can be drawn as dots farther beyond the "whiskers" of the box plot

Practice: Box Plots

I randomly sample 8 wallets to see how much cash is inside each of them. Here are the results:

5 15 25 25 40 65 85 200

Find the first quartile Q1, the median Q2, and the third quartile Q3.

Practice: Box Plot

House prices (in thousands) for two neighborhoods, Twin Pines and Sadlands, are shown in boxplots.


Five number summary:


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