1.4 Measuring Position of Quantitative Data

Original textbook reading. The original text, captured examples and choices appear below. Follow the Fall 2026 course outline for current topics, lessons and complete student checks.

This preserved reading describes percentile rank using observations strictly below a value. The current Topic 1.7 lesson and checks explicitly use the empirical percentage at or below a value, including equal observations. These conventions can give different results, particularly with ties; state the convention and keep the comparison consistent. To report the current empirical percentage, divide the number of observations at or below the value by the total and multiply by 100. Use the current lesson, check and separate worked answer for that convention.

Summary Statistics: Calculate and Choose

Practice the inclusive percentile comparison · Separate worked answer

Section Overview

Interactive 1.4.1 - Cum. Rel. Freq. Graph
Figure 1.4.1
Example 1.4.1
Interactive 1.4.2 - Example 1.4.1
Pop Quiz 1.4.1

Percentiles

One way to measure the position of an observation is to find the pth percentile of the distribution, or the observation with p percent of the observations lesser than it. We’ve actually been learning about percentiles since the last section without even knowing it, as the values of the first quartile, the median, and the third quartile are also the 25th, 50th, and 75th percentiles respectively. We can calculate the percentile of an observation as:

\text{Percentile}=\frac{\text{Number of Lesser Observations}}{\text{Number of Total Observations}}

If we want to display percentiles, we can create a cumulative relative frequency graphGrouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.. An example of one can be seen in Interactive 1.4.1 Before we can create the graph, it helps to first create a cumulative frequency tableGrouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties. and/or a cumulative relative frequency table. The cumulative frequency tables are similar to the frequency tables and relative frequency tables we’ve talked about in previous sections, but instead of just listing the counts or percents for each class, we add the counts or percents from each previous class to the current class as we go down the class list. To better help explain these concepts, let’s look at the following master table, which uses the data from Example 1.2.3, the toy collection problem, as well as includes the cumulative frequency and cumulative relative frequency.

Toys Collected	Freq.	Rel. Freq.	Cumul. Freq.	Cumul. Rel. Freq.
0 to <5	17	43.6%	17	43.6%
5 to <10	12	30.8%	29	74.4%
10 to <15	7	17.9%	36	92.3%
15 to <20	2	5.1%	38	97.4%
20 to <25	0	0.0%	38	97.4%
25 to <30	1	2.6%	39	100.0%

Interactive 1.4.1 Cumulative Relative Frequency Graph

Open full-size image S0062

Source gap (S0062): The source graph clips the vertical-axis labels and its right edge. Its annotation labels are shown without opened annotation panels. The accompanying source table supplies the complete cumulative values.

Transforming Data

In addition to percentiles, we can also describe the position of an observation within a distribution in terms of the z-score, which is a standardized value obtained by transforming our observation, namely by subtracting the mean from the value of the observation and dividing by the standard deviation:

z=\frac{x-\bar{x}}{s_x}

The resulting score is measured in the number of standard deviations away from the mean the observation is. Z-scores are especially useful to us because the standardized score we get allows us to compare the position of observations within different distributions, as long as we know the mean and standard deviation for each distribution.

Another way we can transform our data is by adding/subtracting or multiplying/dividing by a constant. This can prove especially useful when we want to convert our units to another form, such as adding by 273 to convert from degrees Celsius to degrees Kelvin or multiplying by 1.65 to convert from a British Pound to a US Dollar (of course the exact rate of exchange depends on the international economy!) By converting the data, however, we risk changing the center, location, and/or the spread of our distribution. Figure 1.4.1 provides a reference as to which measures are changed depending on if we add by a constant a, multiply by a constant b, or both, where x represents all observations within a distribution:

Figure 1.4.1

Figure 1.4.1 — transformation of the distribution’s shape, center, location, and spread
Distribution	Shape	Center	Location	Spread
x (represents all observations in distribution)	Shape of overall distribution	$\bar{x}$ (mean) M (median)	p (value at pth percentile) $Q_1;\ Q_3$ (1st & 3rd quartile)	IQR (Interquartile Range) $S_x$ (Std. Deviation) Range
$a+x$	same as x	$a+\bar{x}$ $a+M$	$a+p$ $a+Q_1$ $a+Q_3$	same as x
$bx$	same as x	$b\bar{x}$ $bM$	$bp$ $bQ_1$ $bQ_3$	$\|b\|\,IQR$ $\|b\|\,S_x$ $\|b\|\,\text{range}$
$a+bx$	same as x	$a+b\bar{x}$ $a+bM$	$a+bp$ $a+bQ_1$ $a+bQ_3$	$\|b\|\,IQR$ $\|b\|\,S_x$ $\|b\|\,\text{range}$

Source gap (S0064): Source scope: the transformation table states “same as x” for shape and gives the displayed percentile and quartile transformations without stating a restriction on the sign of b. The table is transcribed as shown.

Example 1.4.1

From Example 1.3.1, recall that you, a recent college graduate, were given the bimonthly temperature readings of two cities where you received job offers:

Month	Rusticville Temperatures (°C)	Urbanopolis Temperatures (°C)
January	−12	−4
January	−11	−3
February	−10	−1
February	−9	0
March	−6	6
March	−5	7
April	−2	12
April	−1	15
May	0	17
May	1	20
June	9	28
June	11	29
July	18	36
July	24	39
August	4	25
August	6	26
September	2	21
September	3	24
October	−4	9
October	−3	11
November	−8	3
November	−7	4
December	−16	−9
December	−14	−6

Using what we’ve learned in this section:

a) Find the percentile for the −7°C observation in the Rusticville distribution.

b) Compare the z-scores for the largest temperatures in each of the distributions.

c) Convert the temperatures from degrees Celsius to degrees Fahrenheit for the Rusticville distribution (Hint: $\frac{9}{5}C+32=F$ ). What are the new measures of shape, center, location, and spread?

Example 1.4.1 — recorded solution

a) If the observations are ordered from least to greatest, we can see that the −7°C observation is larger than 7 other observations. Using the percentile formula:

\text{Percentile}=\frac{\text{Number of Lesser Observations}}{\text{Number of Total Observations}}

\frac{7}{24}\approx0.29

The −7°C observation is approximately the 29th percentile of temperatures in the Rusticville distribution

b) When we ordered both of our distributions from least to greatest in Example 1.3.1, we see that the maximum values are 24°C for the Rusticville distribution and 39°C for the Urbanopolis distribution.

Now, we’ll need the mean and standard deviation for each distribution to calculate the individual z-scores.

For the Rusticville distribution, $\bar{x}=-1.25^\circ\mathrm{C}$ and $s_x=9.852^\circ\mathrm{C}$ , so our z-score is:

z=\frac{x-\bar{x}}{s_x}=\frac{24-(-1.25)}{9.852}=2.563^\circ\mathrm{C}

For the Urbanopolis distribution, $\bar{x}=12.875$ and $s_x=13.665$ :

z=\frac{x-\bar{x}}{s_x}=\frac{39-12.875}{13.665}=1.912^\circ\mathrm{C}

The 24°C and 39°C were the largest observations in their respective distributions, but the z-scores are not the same; the z-score for 24°C, 2.563°C is greater than the z-score for 39°C, 1.912°C. This actually makes a lot of sense, as 24°C was proven in Example 1.3.1 to be an outlier in the Rusticville distribution while 39°C was not an outlier in the Urbanopolis distribution.

c) Converting from degrees Celsius to degrees Fahrenheit has us both adding a constant ( $a=32$ ) and multiplying by a constant ( $b=\frac95$ ). Looking at Figure 1.4.1, we can see how this effects the shape, center, location, and spread of the Rusticville distribution:

Shape: The shape of the distribution remains the same even after multiplying and adding by a constant.

Center: Our mean becomes

\frac95\cdot(-1.25)+32=29.75^\circ\mathrm{F},

and our median becomes

\frac95\cdot(-2.5)+32=27.5^\circ\mathrm{F}

Location: In part (a), we found the −7°C observation to be in the 29th percentile. If this observation were converted to Fahrenheit, its new value would be

\frac95\cdot(-7)+32=19.4^\circ\mathrm{F}.

Likewise, our first quartile becomes

\frac95\cdot(-8.5)+32=16.7^\circ\mathrm{F}.

Our third quartile becomes

\frac95\cdot(3.5)+32=38.3^\circ\mathrm{F}.

Spread: Using the values we just found for the first and third quartile, our IQR becomes

38.3-16.7=21.6^\circ\mathrm{F}.

Our standard deviation becomes

\frac95(9.852)=17.734^\circ\mathrm{F}.

Source gap (S0066): Source units: the recorded z-score calculations and comparison append “°C” to 2.563 and 1.912. The section’s definition describes z-scores as numbers of standard deviations. The recorded units are retained without correction.

Interactive 1.4.2 Example 1.4.1

Open full-size image S0068

Source gap (S0068): The captured graph clips the Celsius plot’s left side and the Fahrenheit plot’s right endpoint. The Celsius maximum is labeled “20”, which differs from the Rusticville table’s maximum 24 and the preceding boxplot’s largest nonoutlier 18. Its two annotation labels are visible without opened panel text.

Pop Quiz 1.4.1

Question 1 of 5

For the following cumulative relative frequency table, what is the most likely shape for the distribution?

Classes	Cum. Rel. Frequency
0 to < 4	9%
4 to < 8	17%
8 to < 12	22%
12 to < 16	20%
16 to < 20	32%

A. Left SkewedB. Right SkewedC. SymmetricD. None of the above

Source gap (S0069): Source table labeling is retained: the column is headed “Cum. Rel. Frequency”, but its displayed values decrease from 22% to 20% and total 100% across rows. No alternative table or checked answer is captured.

Question 2 of 5

The following 10 observations are test scores for a Biostatistics course: 64, 67, 73, 76, 81, 81, 83, 88, 92, 96. At what percentile is the student who got the second 81?

A. 40%B. 45%C. 60%D. 81%

Question 3 of 5

Which of the following cumulative relative frequency graphs represents a symmetric distribution?

Open full-size image S0071

Horizontal value	Upper left	Upper right	Lower left	Lower right
0	0	0	0	0
3	0.3	0.2	0.2	0.1
6	0.4	0.6	0.4	0.3
9	0.7	0.6	0.6	0.7
12	0.8	0.2	0.8	0.9
15	1	0	1	1

A. Upper left graphB. Upper right graphC. Lower left graphD. Lower right graph

Question 4 of 5

Which of the following measures are resistant?

A. z-scoreB. PercentileC. Both A and BD. None of the Above

Question 5 of 5

If 10 and 20 are values in a distribution and they increase to 25 and 40 respectively, the values of the constants a and b are:

A. a = 2.5, b = 1B. a = 2, b = 5C. a = 5, b = 2D. a = 2, b = 10

Source gap (S0069): The five quiz questions are captured with no checked-answer or feedback state. Question 3’s graph choices have no letter labels in the source, so their positions identify the reconstructed choices. No source answer key is available.

Glossary and deeper reading · 7 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Observation
An observation is a recorded value or set of values for an individual or occasion. The individual may be a person, object, or event rather than a human subject. For example, one plant’s measured height is an observation in a plant-growth study.
Read more: Observation in the captured Statistics reading →
Percentile
A percentile identifies a location in an ordered distribution by the proportion below or at that location. Ties and finite samples require a stated convention, so methods can produce slightly different percentile values. For example, the 80th percentile marks a value with roughly 80% of observations at or below it.
Read more: Percentile in the current summary-statistics lesson →
Cumulative relative frequency
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Cumulative relative frequency adds proportions through a specified value or interval. A cumulative relative frequency graph shows how the accumulated proportion rises from 0 toward 1. For example, a value of 0.7 at 20 minutes means that 70% of the recorded waits are at or below 20 minutes.
Cumulative frequency
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Cumulative frequency adds counts through a specified value or interval. It shows how many observations lie at or below that point in the ordered data. For example, adding counts for all intervals up to 20 minutes gives the number of recorded waits no longer than 20 minutes.
Z-score
A z-score expresses a value’s distance from its distribution’s mean in standard-deviation units. It is found by subtracting the mean from the value and dividing by the standard deviation. For example, a z-score of −1.5 places the value 1.5 standard deviations below the mean.
Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
Read more: Mean in the captured Statistics reading →
Standard deviation
Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.
Read more: Standard deviation in the captured Statistics reading →

Horizontal value	Upper left	Upper right	Lower left	Lower right
0	0	0	0	0
3	0.3	0.2	0.2	0.1
6	0.4	0.6	0.4	0.3
9	0.7	0.6	0.6	0.7
12	0.8	0.2	0.8	0.9
15	1	0	1	1

Horizontal value	Upper left	Upper right	Lower left	Lower right
0	0	0	0	0
3	0.3	0.2	0.2	0.1
6	0.4	0.6	0.4	0.3
9	0.7	0.6	0.6	0.7
12	0.8	0.2	0.8	0.9
15	1	0	1	1

Section Overview

Percentiles

Interactive 1.4.1 Cumulative Relative Frequency Graph

Transforming Data

Figure 1.4.1

Interactive 1.4.2 Example 1.4.1

Pop Quiz 1.4.1

Observation

Percentile

Cumulative relative frequency

Cumulative frequency

Z-score

Mean

Standard deviation

Help us improve Omniguides

Horizontal value	Upper left	Upper right	Lower left	Lower right
0	0	0	0	0
3	0.3	0.2	0.2	0.1
6	0.4	0.6	0.4	0.3
9	0.7	0.6	0.6	0.7
12	0.8	0.2	0.8	0.9
15	1	0	1	1