Unit 3 Statistics: Five-Number Summaries, Box Plots, and Outlier Analysis

Learning Intent and Success Criteria

  • Unit 3: Statistics

  • Topic: Box plots

  • Timeline: Week 2 – Lesson 2

Learning Objectives: I am successful when I can:

  • Calculate a 5-number summary.

  • Construct a box plot.

  • Identify outliers.

  • Interpret results.

  • State a 5-number summary correctly.

  • Organise the 5-number summary into a box plot.

  • Recognise and perform a test for outliers.

  • Analyse a box plot.

Analysing Data - Measures of Spread

  • Measures of spread describe the variability in a sample or population.

  • They indicate how wide the data set is.

  • Two common measures of spread include:

    1. The range

    2. The interquartile range

The Range

  • Definition: The range of a data set is the difference between the smallest (minimum) and largest (maximum) values in the data set.

  • Formula: Range=maximum value−minimum value\text{Range} = \text{maximum value} - \text{minimum value}

Worked Example:

  • Data set: $23, 25, 45, 48, 52, 55, 59, 62, 85$

  • Calculation: Range=85−23=23\text{Range} = 85 - 23 = 23

Quartiles

  • Definition: Quartiles are the values that divide an ordered data set into four (approximately) equal parts.

  • There are three quartiles:

    • The first/lower quartile (Q1Q_1): The value that divides approximately the lower 25%25\% of data values from the upper 75%75\% of data values.

    • The second quartile (Q2Q_2): This is the median of the data set.

    • The third/upper quartile (Q3Q_3): The value that divides approximately the upper 25%25\% of data values from the lower 75%75\% of data values.

  • Distribution: Each quartile contains 25%25\% of the data set scores.

Finding Quartiles (Odd vs. Even Number of Scores):

  1. Put the numbers in order.

  2. Locate the median then separate the values below it from the values above it.

  3. Find Q1Q_1 (the median of the lower half) and Q3Q_3 (the median of the upper half).

Examples:

  • Sorted distribution: $1, (2), 2, 3, 4, 5, 5$

    • Q1Q_1 (Lower quartile) = $2$

    • Q2Q_2 (Median)

    • Q3Q_3 (Upper quartile)

  • Other examples provided result in:

    • Q1=57Q_1 = 57

    • Q3=81Q_3 = 81

Interquartile Range (IQR)

  • Definition: The interquartile range (IQR) is a measure of the spread within a numerical data set.

  • The IQR consists of the middle 50%50\% (approximately) of the data values.

  • Formula: IQR=Q3−Q1\text{IQR} = Q_3 - Q_1

Worked Example:

  • Data set: $141, 153, 158, 161, 163, 164, 166, 168, 168, 173, 174, 195$

  • Calculated Range: Range=195−141=54 cm\text{Range} = 195 - 141 = 54\,\text{cm}

  • Example calculation: $1, 2, 3, 3, 5, 6, 6, 7, 8$

    • Range=highest−lowest\text{Range} = \text{highest} - \text{lowest}

Practical Application: Year 10 Income Data

Scenario: Some Year 10 students were asked to record how much money they received in a particular week (pocket money or part-time work). Data for 20 students (n=20n=20): $0, 0, 0, 0, 0, 0, 0, 5, 10, 13, 15, 20, 20, 25, 30, 30, 40, 79, 80, 260$

Tasks:

  1. Find the range.

  2. Find the interquartile range.

  3. Comment on the answers (e.g., impact of the $260 extreme value).

Five-Number Summary

  • A five-number summary is written in the following order: {min,Q1,Q2,Q3,max}\lbrace \text{min}, Q_1, Q_2, Q_3, \text{max} \rbrace

Worked Example:

  • Data: $22, 30, 32, 35, 35, 39, 41, 45, 51, 60$

  • Median (Q2Q_2): Middle of $35$ and $39 = \frac{35+39}{2} = 37$

  • Q1Q_1 (Middle of bottom half): $32$

  • Q3Q_3 (Middle of top half): $45$

  • Five-number summary: {22,32,37,45,60}\lbrace 22, 32, 37, 45, 60 \rbrace

Box Plots

  • Definition: A box plot is a graphical representation of a five-number summary.

  • Components of a Box Plot:

    • Title: e.g., "Ages of swimming club members".

    • Scale: A numerical axis covering the full range of data.

    • Scale identifier: Labels the units (e.g., "Age in years").

    • Whiskers: Lines extending from the box to the minimum and maximum values.

    • Box: Represents the range from Q1Q_1 to Q3Q_3.

    • Median Line: A vertical line inside the box at Q2Q_2.

Vertical Box Plots Worked Example:

  • Summary: min=1,Q1=3,Q2=4,Q3=5,min=7\text{min} = 1, Q_1 = 3, Q_2 = 4, Q_3 = 5, \text{min} = 7 (Note: Max is 7).

  • Steps:

    1. Set up a horizontal or vertical number line to cover the range from $1$ to $7$.

    2. Locate max ($7$) and min ($1$) adjacent to the scale.

    3. Locate Q1Q_1, Q2Q_2, Q3Q_3. Draw horizontal and vertical lines to create the box.

    4. Draw in the whiskers and end lines.

Practice Exercises: Box Plot Construction

Exercise 1:

  • Distribution: $1, 4, 2, 5, 6, 2, 3, 7, 5$

  • Sorted list: $1, 2, 2, 3, 4, 5, 5, 6, 7$

  • Five-number summary: {1,2,4,5.5,7}\lbrace 1, 2, 4, 5.5, 7 \rbrace

Exercise 2:

  • Distribution: $23, 24, 25, 25, 26, 27, 27, 27, 28, 28, 28, 28, 28, 29, 30$

Exercise 3:

  • Data: $15, 12, 20, 11, 15, 13, 9, 13, 19, 16, 6, 10, 17, 11, 9$

Box Plot Shape and Interpretation

The shape of a box plot indicates the nature of the data distribution:

  • Symmetrical: The median is in the center of the box, and whiskers are of approximately equal length.

  • Positive Skew: The whisker is longer on the right side (higher values); the median is shifted toward the left side (lower values).

  • Negative Skew: The whisker is longer on the left side (lower values); the median is shifted toward the right side (higher values).

Identifying and Proving Outliers

  • Outlier Definition: A data value that appears to stand out by being unusually high or low.

  • Impact: Extreme values make whiskers appear longer, suggesting the data is more spread out than it truly is.

  • Graphing Outliers: Outliers are denoted by a dot or a small cross ($X$). The whisker is then shortened to the next smallest or largest actual score in the data set.

Proving an Outlier Exists (Mathematical Test):

  • Data set: $1, 1, 2, 3, 4, 5, 6, 13$

  • Summary statistics: Q1=1.5Q_1 = 1.5, Q3=5.5Q_3 = 5.5

  • IQR=Q3−Q1=5.5−1.5=4\text{IQR} = Q_3 - Q_1 = 5.5 - 1.5 = 4

  • Upper Limit Test: Upper Bound=Q3+1.5×IQR\text{Upper Bound} = Q_3 + 1.5 \times \text{IQR}

  • Calculation: 5.5+1.5×4=5.5+6=11.55.5 + 1.5 \times 4 = 5.5 + 6 = 11.5

  • Conclusion: Since $13 > 11.5$, the value $13$ is a proven outlier.

Step-by-Step Outlier Identification Example: Data: $12, 20, 17, 10, 17, 18, 14, 29, 14, 10, 11, 12, 11, 9, 20, 17$

  1. Sort the data: $9, 10, 10, 11, 11, 12, 12, 14, 14, 17, 17, 17, 18, 20, 20, 29$

  2. Identify unusual scores: $29$ is unusually high.

  3. Five-number summary: min=9,Q1=11,Q2=14,Q3=17.5,max=29\text{min} = 9, Q_1 = 11, Q_2 = 14, Q_3 = 17.5, \text{max} = 29

  4. Action: Recognize the long upper whisker. Mark $29$ as an $X$ and redraw the whisker to the next score ($20$).

Questions & Discussion

Question Set 1: Box Plot Statistics

  • What numbers represent statistics in the five-number summary?

  • What does the box represent?

  • What measure of centre is not represented on a box plot? (Answer: The Mean).

Question Set 2: Thai Takeaway Waiting Times

  • a) State the median waiting time and explain its significance.

  • b) What is the range of this set of data?

  • c) What is the interquartile range of this set of data?

  • d) What does the interquartile range tell you about the data?

Question Set 3: Health Club Weigh-in

  • a) What is the median of this data?

  • b) What is the range of this data?

  • c) Between what two values does the middle half of the data lie?

  • d) What percentage of people in the group weigh less than $70\,\text{kg}$?

Textbook Exercises (Ex 12.3 & 12.4)

Individual Pathways - Ex 12.3:

  • Practise: $1, 4, 7, 10, 13$

  • Consolidate: $2, 6, 8, 11, 14$

  • Master: $3, 5, 9, 12, 15$

Individual Pathways - Ex 12.4:

  • Practise: $1, 4, 6, 9, 12, 13, 16, 19$

  • Consolidate: $2, 5, 7, 10, 14, 17, 20$

  • Master: $3, 8, 11, 15, 18, 21, 22$

Extension: Standard Deviation

Refer to Section 12.5 (Page 898–906) and Ex 12.5. Step-by-Step Process to Calculate Standard Deviation:

  1. Calculate the mean.

  2. Calculate the deviations (difference between each value and the mean).

  3. Square each deviation.

  4. Sum the squares.

  5. Divide the sum of the squares by the total number of data values.

  6. Take the square root of the result.