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1.10 · Use measures of position and box plots

Learn to use measures of position and box plots through clear examples and targeted practice.

Athabasca University MATH 215: Introduction to Statistics

Descriptive Statistics

Describing where values lie in a data set

A data value is easier to interpret when you know where it sits among the other values. A travel time of 20 minutes, for example, may be short in one group and long in another. Measures of position describe relative location, and a box plot gives a visual summary of location and spread. This lesson covers z-scores, quartiles, and box plots using introductory methods. For ordered-data calculations, first sort the values from smallest to largest. This is the same basic idea as arranging numbers in order before identifying a middle value.

What you will learn

1. Measures of position: z-scores

A measure of position describes where a value lies in relation to other values or to a distribution. A z-score measures how many standard deviations a value is above or below the mean. The mean is the arithmetic average. The standard deviation describes the typical distance of values from the mean.
For a value xx, population mean μ\mu, and population standard deviation σ\sigma, use the z-score formula. If you are describing a sample and have its sample mean xˉ\bar{x} and sample standard deviation ss, use those sample quantities. In either case, subtract the mean from the value and divide by the standard deviation. The units cancel, so a z-score has no units.
Before calculating, check that the value, mean, and standard deviation refer to the same variable and use the same units. The standard deviation must be positive. A positive z-score means the value is above the mean; a negative one means it is below. A z-score describes distance from the mean in standard-deviation units. It does not, by itself, tell you what percentage of observations are above or below the value.
z=x−μσz=\frac{x-\mu}{\sigma}

2. Quartiles and the interquartile range

Quartiles mark positions in an ordered data set. The first quartile, Q1Q_1, marks the lower-quarter position; the second quartile, Q2Q_2, is the median; and the third quartile, Q3Q_3, marks the upper-quarter position. The median is the middle value after sorting. If there are an even number of values, find it by averaging the two middle values.
There is more than one convention for finding quartiles, particularly when the number of observations is odd. To make the calculations in this lesson consistent, use the median-of-halves rule. Find the overall median, then find the median of the lower half and the median of the upper half. If the number of values is odd, leave the overall median out of both halves. If it is even, divide the ordered values into two equal halves.
The interquartile range, abbreviated IQR, is the difference between the third and first quartiles. It describes the spread of the middle half of the observations and uses the same units as the data. The five-number summary consists of the minimum, first quartile, median, third quartile, and maximum. A summary should identify its units and the quartile convention when that convention matters.
IQR=Q3−Q1\mathrm{IQR}=Q_3-Q_1

3. Building and reading a box plot

A box plot, also called a box-and-whisker plot, displays a five-number summary on a number line. Draw the box from Q1Q_1 to Q3Q_3 and draw a line inside it at the median. The box's length represents the IQR. Whiskers show how far the data extend beyond the box under the plotting convention being used.
One common introductory convention uses the 1.5-IQR rule to flag potential outliers. First calculate the lower and upper fences using the formula. Observations below the lower fence or above the upper fence are flagged. The fences are boundaries for identifying potential outliers, not observations themselves. A whisker should not end at a fence unless a data value is actually there.
For a box plot using this convention, whiskers extend from the box to the smallest and largest observations that are not flagged. Plot flagged observations separately beyond the whiskers. Other box plots may use different whisker conventions, so read any accompanying description. A box plot helps compare centre and spread, but it does not display every observation or reveal every detail of the data.
L=Q1−1.5 IQR,U=Q3+1.5 IQRL=Q_1-1.5\,\mathrm{IQR},\quad U=Q_3+1.5\,\mathrm{IQR}

4. Interpreting position in context

A population is the full group of interest. A sample is the subset of that group whose data are observed. A variable is the characteristic recorded for each person or item, such as travel time in minutes. A parameter is a numerical summary of a population; a statistic is a numerical summary calculated from a sample. These terms clarify which group a result describes.
When interpreting a box plot, name the variable and units, then describe the median and IQR in context. A higher median indicates a higher middle value. A larger IQR indicates more spread among the middle half of the observations. If the median is closer to one edge of the box, or one whisker is longer than the other, the data may be less balanced around the centre. Describe what the plot shows without claiming more than it supports.
A z-score and a box plot describe position in different ways. A z-score locates one value relative to a mean and standard deviation. A box plot summarizes ordered data using quartiles and the range. Use the reference group named in the problem, and include units when interpreting measurements.

Worked example

Quartiles, fences, and a box plot

Nine students report their travel times to campus, in minutes: 8, 12, 14, 15, 16, 18, 20, 23, and 40. Find the five-number summary and IQR, check for potential outliers using the 1.5-IQR rule, and describe the box plot.
  1. Identify the data and convention
    The variable is travel time in minutes for these nine students. The observations are already ordered from smallest to largest. Use the median-of-halves rule, leaving the overall median out of both halves because the number of observations is odd.
  2. Find the median and quartiles
    The fifth value is the overall median, Q2Q_2. The lower half is 8, 12, 14, and 15; the upper half is 18, 20, 23, and 40. Each half has four values, so its median is the average of its two middle values.
    Q1=12+142=13,Q2=16,Q3=20+232=21.5Q_1=\frac{12+14}{2}=13,\quad Q_2=16,\quad Q_3=\frac{20+23}{2}=21.5
  3. Calculate the IQR
    Subtract the first quartile from the third quartile. The result is in minutes and describes the spread of the middle half of these travel times.
    IQR=21.5−13=8.5 min\mathrm{IQR}=21.5-13=8.5\text{ min}
  4. Calculate and check the fences
    Use the IQR to find the lower and upper fences. Compare the observed minimum and maximum with the boundaries to check whether either is flagged as a potential outlier.
    L=13−1.5(8.5)=0.25,U=21.5+1.5(8.5)=34.25L=13-1.5(8.5)=0.25,\quad U=21.5+1.5(8.5)=34.25
  5. Describe the box plot
    The minimum, 8 minutes, is above the lower fence and is not flagged. The maximum, 40 minutes, is above the upper fence and is flagged. Draw the box from 13 to 21.5 minutes, with its median line at 16. The upper whisker reaches 23 minutes, the largest unflagged value; plot the 40-minute observation separately.
Answer: The five-number summary is: minimum 8 minutes, Q1=13Q_1=13 minutes, median 16 minutes, Q3=21.5Q_3=21.5 minutes, and maximum 40 minutes. The IQR is 8.5 minutes. The 40-minute travel time is a potential outlier under the 1.5-IQR rule. The box plot shows that the middle half of these travel times spans 8.5 minutes.
Check: The calculations use the stated median-of-halves convention. Since 40 minutes is greater than the upper fence of 34.25 minutes, it is correctly flagged.

Common mistakes and how to avoid them

Finding quartiles before sorting the observations.
Correction: Put the data in increasing order first. Quartiles depend on positions in the ordered list.
Including the overall median in both halves when using the median-of-halves rule for an odd number of observations.
Correction: For the convention used here, leave the overall median out of both halves when the number of values is odd.
Calling every flagged value an error or removing it automatically.
Correction: The fence rule identifies potential outliers. Check the context and data quality before deciding how to handle a value.
Saying that the IQR measures the full range of the data.
Correction: The IQR measures the distance from Q1Q_1 to Q3Q_3; the full range runs from the minimum to the maximum.

Lesson summary

Check your understanding

Question 1

A value is 6 units above a mean, and the standard deviation is 3 units. What is its z-score?
  1. 2
  2. -2
  3. 0.5
  4. 9
Show answer and explanation
2
Subtract the mean from the value to get 6, then divide by the standard deviation: 6/3=26/3=2. The value is two standard deviations above the mean.

Question 2

For a data set, Q1=10Q_1=10 and Q3=18Q_3=18. What is the IQR?
  1. 8
  2. 14
  3. 28
  4. 4
Show answer and explanation
8
The IQR is the third quartile minus the first quartile: 18−10=818-10=8.

Question 3

If the median line is close to the left edge of a box, what is the safest conclusion?
  1. The median lies close to the first quartile compared with the box width.
  2. The mean must equal the median.
  3. Every value is evenly spaced.
  4. The data contain no outliers.
Show answer and explanation
The median lies close to the first quartile compared with the box width.
The median line's position shows that the median is relatively close to Q1Q_1 within the middle half. The box plot alone does not establish the other claims.

Key terms

Measure of position
A numerical or visual description of where a value lies relative to other values or a distribution.
Z-score
The number of standard deviations a value lies above or below the mean.
Quartile
One of three values used to mark quarter positions in an ordered data set.
Interquartile range
The difference between the third and first quartiles, describing the spread of the middle half.
Five-number summary
The minimum, first quartile, median, third quartile, and maximum of a data set.
Box plot
A graph that displays quartiles, the median, and whiskers to summarize a distribution.
Potential outlier
An observation beyond a specified boundary, such as a fence from the 1.5-IQR rule, that may be unusual and merits attention.

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Published by DoAssignment. This AI-assisted lesson follows Athabasca University MATH 215: Introduction to Statistics, study topic 1.10. It is a study resource, not an official curriculum publication.

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