3.4: Measures of Position

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Toros Berberyan, Tracy Nguyen, and Alfie Swan
Citrus College

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Learning Objectives

Identify the position of data values within a data set using measures of position.
Calculate z-scores to determine how many standard deviations a value is from the mean.
Use percentiles and quartiles to divide data into ranked sections.

Measures of position help describe where a specific data point stands about the entire dataset. Z-scores measure how many standard deviations a value is from the mean, helping identify outliers and standardize comparisons across different datasets. Percentiles indicate the percentage of values below a given data point, commonly used in standardized testing and growth charts. Quartiles divide data into four equal parts, with Q1 (25th percentile) representing the lower quarter, Q2 (50th percentile or median) marking the middle, and Q3 (75th percentile) showing the upper quarter. These measures are essential for understanding data distribution, detecting outliers, and making informed statistical decisions.

Z-scores

A z-score tells you how far a data point is from the average (mean) of a data set, measured in standard deviations. It helps compare values from different distributions by showing if a number is above or below the average and by how much. Z-scores help standardize values so they can be compared, even if they come from different groups or scales.

Definition: Z-Score Formula

$Z =\dfrac{X - \bar{X}}{s}$

Meaning of Each Variable:

$ Z $ represents the z-score (standardized value), which tells how many standard deviations a data point is from the sample mean
$ X $ represents the individual data point (observed value)
$ \bar{X} $ represents the sample mean (average of the sample data)
$ s $ represents the sample standard deviation

Example $\PageIndex{1}$

A professor in a Psychology 101 class gives a test, and the students' scores are recorded. Suppose the sample mean test score is 78 with a sample standard deviation of 10.

Two students, Alex and Jordan, received the following scores:

Alex's score: 85
Jordan's score: 65

Solution

Using the z-score formula:

$Z= \dfrac{X - \bar{X}}{s}$

where:

$X$ is the individual test score
$\bar{X}=78$ (sample mean)
$s=10$ (sample standard deviation)

Now, let's compute their z-scores.

Alex's Z-Score:

$Z=\dfrac{85 - 78}{10} = \dfrac{7}{10} = 0.7$

Jordan's Z-Score:

$Z = \dfrac{65 - 78}{10} = \dfrac{-13}{10} =−1.3$

Example $\PageIndex{2}$

A server at a restaurant tracks their weekly tips earned and hours worked. Over several weeks, the average weekly tips earned is $500 with a standard deviation of $100. This week, the server earned $650 in tips. Meanwhile, the average number of hours worked per week is 30 hours with a standard deviation of 5 hours, and this week, the server worked 40 hours.

Using z-scores, determine whether the server's weekly tips or weekly hours worked deviate more from the average, indicating where their performance was stronger relative to past trends.

Solution

Using the z-score formula:

Z-score for weekly tips: $Z = \dfrac{650 - 500}{100} = \dfrac{150}{100} = 1.5$
Z-score for weekly hours worked: $Z = \dfrac{40 - 30}{5} = \dfrac{10}{5} = 2.0 $

Conclusion:

The z-score for hours worked 2 is higher than the z-score for tips earned 1.5. This means that relative to their average, the server worked significantly more hours than usual compared to how much more they earned in tips. Therefore, their performance in terms of hours worked deviated more from the average than their tips.

Percentiles

A percentile is a measure of ranking. It represents a location measurement of a data value to the rest of the values. Many standardized tests give the results as a percentile. Doctors also use percentiles to track a child’s growth.

The kth percentile is the data value that has k% of the data at or below that value.

Example $\PageIndex{3}$ interpreting percentile

What does a score of the 90th percentile mean?
What does a score of the 70th percentile mean?

Solution

This means that 90% of the scores were at or below this score. (A person did the same as or better than 90% of the test takers.)
This means that 70% of the scores were at or below this score.

Example $\PageIndex{4}$ percentile versus score

If the test was out of 100 points and you scored at the 80^th percentile, what was your score on the test?

Solution

You don’t know! All you know is that you scored the same as or better than 80% of the people who took the test. If all the scores were low, you could have still failed the test. On the other hand, if many of the scores were high you could have gotten a 95% or so.

Quartiles

Quartiles are a way of splitting a group of numbers into four equal parts to help understand how data is spread out. Imagine you line up all your test scores from lowest to highest. Quartiles break this list into four sections:

First quartile (Q₁): The number that separates the lowest 25% of scores from the rest.
Second quartile (Q₂ or median): The middle number that splits the data into two halves.
Third quartile (Q₃): The number that separates the highest 25% of scores from the rest.

Why are quartiles useful?

Understanding data spread – They help show if data is grouped closely or spread out.
Finding outliers – If a value is far beyond Q1 or Q3, it might be an unusual (outlier) data point.
Comparing groups – They help compare data sets, like test scores of different classes.
Making box plots – Quartiles are used to create box-and-whisker plots, a great way to visualize data distribution.

Simply put, quartiles help break large amounts of information into smaller, meaningful parts, making it easier to analyze and compare.

Definition $\PageIndex{1}$

To find the quartiles:

Sort the data in increasing order.
Find the median, this divides the data list into 2 halves.
Find the median of the data below the median. This value is Q₁.
Find the median of the data above the median. This value is Q₃.
Ignore the median in both calculations for Q₁ and Q₃.

If you record the quartiles together with the maximum and minimum you have five numbers. This is known as the five-number summary. The five-number summary consists of the minimum, the first quartile (Q₁), the median, the third quartile (Q₃), and the maximum (in that order).

The interquartile range, IQR, is the difference between the first and third quartiles, Q₁ and Q₃. Half of the data (50%) falls in the interquartile range. If the IQR is “large” the data is spread out and if the IQR is “small” the data is closer together.

Definition $\PageIndex{2}$

Interquartile Range (IQR)

IQR = Q₃ – Q₁

Determining probable outliers from IQR: fences

A value that is less than Q₁–$1.5 \cdot $IQR (this value is often referred to as a lower fence) is considered an outlier.

Similarly, a value that is more than Q₃$+1.5 \cdot $IQR (the upper fence) is considered an outlier.

A box plot (or box-and-whisker plot) is a graphical display of the five-number summary. It can be drawn vertically or horizontally. The basic format is a box from Q1 to Q3, a vertical line across the box for the median and horizontal lines as whiskers extending out each end to the minimum and maximum. The minimum and maximum can be represented with dots. Don’t forget to label the tick marks on the number line and give the graph a title.

If the data are symmetrical, then the box plot will be visibly symmetrical. If the data distribution has a left skew or a right skew, the line on that side of the box plot will be visibly long. If the plot is symmetrical, and the four quartiles are all about the same length, then the data are likely a near-uniform distribution. If a box plot is symmetrical, and both outside lines are noticeably longer than the Q1 to median and median to Q3 distance, the distribution is then probably bell-shaped.

Typical box plot. — Figure $\PageIndex{1}$: Typical Box Plot

Example $\PageIndex{5}$ five-number summary for an even number of data points

The total assets in billions of Australian dollars (AUD) of Australian banks for the year 2012 are given in Example $\PageIndex{1}$ ("Reserve Bank of," 2013).

Find the five-number summary, and draw a box-and-whiskers plot.
Find the interquartile range (IQR) and determine if there are any outliers in the raw data.

Table $\PageIndex{1}$: Total Assets (in billions of AUD) of Australian Banks
2855	2862	2861	2884	3014	2965
2971	3002	3032	2950	2967	2964

Solution

Find the five-number summary, and draw a box-and-whiskers plot.

Variable: $x =$ total assets of Australian banks

First sort the data.

Table $\PageIndex{2}$: Sorted Data for Total Assets
2855	2861	2862	2884	2950	2964	2965	2967	2971	3002	3014	3032

The minimum is 2855 billion AUD and the maximum is 3032 billion AUD.

There are 12 data points so the median is the average of the 6th and 7th numbers.

Sorted data with median (quartile two).

Figure $\PageIndex{2}$: Sorted Data for Total Assets with Median

To find Q₁, find the median of the first half of the list.

Sorted data with quartile one.

Figure $\PageIndex{3}$:: Finding Q₁

To find Q₃, find the median of the second half of the list.

Sorted data with quartile three.

Figure $\PageIndex{4}$:: Finding Q₃

The five-number summary is (all numbers in billion AUD)

Minimum: 2855

Q₁: 2873

Median: 2964.5

Q₃: 2986.5

Maximum: 3032

You can use the five-number summary to draw the box-and-whiskers plot.

Box plot of total assets of Australian Banks — Figure $\PageIndex{5}$: Box Plot of Total Assets of Australian Banks

The distribution is skewed right because the right tail is longer.

Find the interquartile range (IQR) and determine if there are any outliers in the raw data.

To find the interquartile range, IQR = Q_{3 –}Q₁

IQR = 2986.5 – 2873 = 113.5 billion AUD

This tells you the middle 50% of assets were within 113.5 billion AUD of each other.

Lower Fence = Q_{1 –}$1.5 \cdot $IQR = 2873 – $1.5 \cdot $113.5 = 2873 - 170.25 = 2702.75

Upper Fence = Q₃+$1.5 \cdot $IQR = 2986.5 + $1.5 \cdot $113.5 = 2986.5 + 170.25 = 3156.75


2855	2861	2862	2884	2950	2964	2965	2967	2971	3002	3014	3032

From our arranged data, there is no outlier because there is no number in the raw data below the lower fence, 2702.75, and higher than the upper fence, 3156.75.

Example $\PageIndex{6}$ five-number summary for an odd number of data points

The life expectancy for a person living in one of 11 countries in the region of South East Asia in 2012 is given below ("Life expectancy in," 2013).

Find the five-number summary, and draw a box-and-whiskers plot.
Find the interquartile range (IQR) and determine if there are any outliers in the raw data.

Table $\PageIndex{3}$: Life Expectancy of a Person Living in South-East Asia
70	67	69	65	69	77
65	68	75	74	64

Solution

Find the five-number summary, and draw a box-and-whiskers plot.

Variable: $x =$ life expectancy of a person.

Sort the data first.

Table $\PageIndex{7}$: Sorted Life Expectancies
64	65	65	67	68	69	69	70	74	75	77

The minimum is 64 years and the maximum is 77 years.

There are 11 data points so the median is the 6th number in the list.

Sorted data with median (quartile two).

Figure $\PageIndex{6}$:: Finding the Median of Life Expectancies

To find the Q₁ and Q₃ you need to find the median of the numbers below the median and above the median. The median is not included in either calculation.

Sorted data with quartile one.

Figure $\PageIndex{7}$: Finding Q₁

Sorted data with quartile three.

Figure $\PageIndex{8}$: Finding Q₃

Q₁=65 years and Q₃=74 years

The five-number summary is (in years)

Minimum: 64

Q₁: 65

Median: 69

Q₃: 74

Maximum: 77

Box plot of life expectancy — Figure $\PageIndex{9}$: Box Plot of Life Expectancy

This distribution looks approximately skewed to the right since the whisker is longer on the right. However, it could be considered almost symmetric too since the box looks somewhat symmetric.

Find the interquartile range (IQR) and determine if there are any outliers in the raw data.

To find the interquartile range (IQR)

IQR=Q₃-Q₁=74-65=9 years

The middle 50% of life expectancies are within 9 years.

Lower Fence = Q₁$-1.5 \cdot $IQR = 65 $-1.5 \cdot $9 = 65 - 13.5 = 51.5

Upper Fence = Q₃+$1.5 \cdot $IQR = 74 + $1.5 \cdot $9 = 74 + 13.5 = 87.5

Data For Life Expectancy
64	65	65	67	68	69	69	70	74	75	77

Table $\PageIndex{4}$: Data for Life Expectancy

From our arranged data, there is no outlier because there is no number in the raw data below the lower fence, 51.5, and higher than the upper fence, 87.5.

You can draw 2 box plots side by side (or one above the other) to compare 2 samples. Since you want to compare the two data sets, make sure the box plots are on the same axes. As an example, suppose you look at the box-and-whiskers plot for life expectancy for European countries and Southeast Asian countries.

Box plots of life expectancy of two regions — Figure $\PageIndex{10}$: Box Plots of Life Expectancy of Two Regions

Looking at the box-and-whiskers plot, you will notice that the three quartiles for life expectancy are all higher for the European countries, yet the minimum life expectancy for the European countries is less than that for the Southeast Asian countries. The life expectancy for the European countries appears to be skewed left, while the life expectancies for the Southeast Asian countries appear to be more symmetric. There are of course more qualities that can be compared between the two graphs.

Authors

"3.4: Measures of Position" by Toros Berberyan, Tracy Nguyen, and Alfie Swan is licensed under CC BY-SA 4.0

Attributions

"3.3: Ranking" by Kathryn Kozak is licensed CC BY-SA 4.0

Exercises

Two students, Alex and Jordan, recently took a Biology 101 exam. The class average score on the exam was 75 with a standard deviation of 8.
- Alex scored 88 on the exam.
- Jordan scored 68 on the exam.

Calculate the z-scores for both Alex and Jordan. Which student performed better relative to the class average? Round to two decimal places.

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Malik recently received grades for two different assessments: an essay in English 103 and an online economics exam. In English 103, the class average score was 82 with a standard deviation of 6, and Malik scored 90 on his essay. In his online economics exam, the class average score was 76 with a standard deviation of 8, and Malik scored 85. Using z-scores (rounded to two decimal places), determine in which subject Malik performed better relative to his classmates.

Suppose your child takes a standardized test in mathematics and scores in the 96^th percentile. What does this percentile mean? Can you say your child passed the test? Explain.

Suppose your child is in the 83rdpercentile in height and 24^th percentile in weight. Describe what this tells you about your child’s stature.

Cholesterol levels were collected from patients two days after they had a heart attack. Find the five-number summary.

Table $\PageIndex{5}$: Cholesterol Levels
Data Column #1	Data Column #2	Data Column #3	Data Column #4	Data Column #5	Data Column #6	Data Column #7
270	226	210	142	280	272	160
220	226	242	186	266	206	318

A group of students recorded the amount of time (in minutes) they spent at the gym on the weekend. Find the five-number summary and determine if there are outliers. The recorded times are as follows:

81, 44, 101, 90, 50, 112, 116, 104, 320, 117, 53, 32, 51, 82, 31

The box and whiskers plot represents the ages of family members. Determine the minimum value, Q₁, Median, Q₃, and the maximum value.

Title: Ages of Family Members. We have a boxplot with a left whisker from 3 to 13, a box from 13 to 32 with a midpoint line inside the box at 22, and a right whisker from 32 to 39.

A study was conducted to see the effect of exercise on pulse rate. Male subjects were taken who do not smoke, but do drink. Their pulse rates were measured ("Pulse rates before," 2013). Then they ran in place for one minute and measured their pulse rate again. The graphs of the box-and-whisker plots below were created for the before and after pulse rates. Discuss any conclusions you can make from the graphs.

The description for this graph is located directly above it. — Title: Pulse Rate of Male Subjects Before and After One Minute of Exercise. For before, we have a boxplot with a left whisker from 50 to 60, a box from 60 to 80 with a midpoint line inside the box roughly at 75, and a right whisker from 80 to 145. For after, we have a boxplot with a left whisker from 80 to 90, a box from 90 to 145 with a midpoint line inside the box at 130, and a right whisker from 145 to 170.

Answers

If you are an instructor and want the solutions to all the exercise questions for each section, please email Toros Berberyan.

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Definition: Z-Score Formula

Example \(\PageIndex{1}\)

Solution

Example \(\PageIndex{2}\)

Solution

Conclusion:

Example \(\PageIndex{3}\) interpreting percentile

Solution

Example \(\PageIndex{4}\) percentile versus score

Solution

Definition \(\PageIndex{1}\)

Definition \(\PageIndex{2}\)

IQR = Q₃ – Q₁

Example \(\PageIndex{5}\) five-number summary for an even number of data points

Solution

Example \(\PageIndex{6}\) five-number summary for an odd number of data points

Solution

Learning Objectives

Z-scores

Definition: Z-Score Formula

Example \(\PageIndex{1}\)

Solution

Example \(\PageIndex{2}\)

Solution

Conclusion:

Percentiles

Example \(\PageIndex{3}\) interpreting percentile

Solution

Example \(\PageIndex{4}\) percentile versus score

Solution

Quartiles

Definition \(\PageIndex{1}\)

Definition \(\PageIndex{2}\)

IQR = Q3 – Q1

Example \(\PageIndex{5}\) five-number summary for an even number of data points

Solution

Example \(\PageIndex{6}\) five-number summary for an odd number of data points

Solution

Authors

Attributions

Exercises

IQR = Q₃ – Q₁