Difficulty: Introductory | Prerequisites: Part 1 of these notes (standard deviation, variance, empirical rule) and Section 2.2 (mean, median).
You now know how to measure centre and spread. This section introduces tools for locating individual data values within a distribution. A z-score tells you how many standard deviations a value sits from the mean, which lets you compare observations from completely different data sets on the same scale. Percentiles and quartiles slice the data into ordered chunks so you can say things like "this value is higher than 75% of all observations." Together, these tools lead to the five-number summary, which gives a compact snapshot of an entire distribution.
You need to be comfortable computing means and standard deviations before starting here, so review Part 1 first if those formulas are not yet solid.
A z-score converts any data value into "how many standard deviations from the mean", making it possible to compare values across different scales. Percentiles and quartiles divide ordered data into percentage-based slices, and the five-number summary (Min, Q1, Median, Q3, Max) captures the shape and spread of a distribution in five values. The IQR (Q3 - Q1) measures the spread of the middle 50% and is resistant to outliers.
Z-score (standardised value)
The number of standard deviations a data value falls from the mean: z = (x - x̄) / s. Think of it as a universal ruler that puts every observation on the same scale, regardless of the original units.
Percentile (Pth percentile)
The value below which P% of the data falls when the data are arranged in ascending order. In simple terms, if you scored at the 90th percentile on a test, you did better than 90% of test-takers.
Quartile
The three values that divide ordered data into four equal parts, each containing 25% of the observations.
First quartile (Q1, lower quartile, 25th percentile)
The median of the lower half of the data. 25% of values fall below Q1.
Second quartile (Q2, median, 50th percentile)
The middle value of the ordered data set. 50% of values fall below Q2.
Third quartile (Q3, upper quartile, 75th percentile)
The median of the upper half of the data. 75% of values fall below Q3.
Interquartile range (IQR)
The distance between the first and third quartiles: IQR = Q3 - Q1. It captures the spread of the middle 50% of the data and is resistant to outliers. Think of it as the range, but ignoring the extreme values on both ends.
Five-number summary
The set {Min, Q1, Median, Q3, Max}. It gives a compact picture of where the data starts, where it clusters, and where it ends.
Range
Maximum - Minimum. The simplest measure of spread, but easily distorted by a single extreme value.
The z-score (standardised value) converts a raw data value into the number of standard deviations it sits from the mean.
For a sample: z = (x - x̄) / s
For a population: z = (x - μ) / σ
Key properties:
Z-scores are unitless. Whether the original data is in inches, bpm, or batting averages, the z-score strips the units away.
A positive z-score means the value is above the mean. A negative z-score means it is below.
The magnitude tells you how far: z = 2 means the value is two standard deviations above the mean.
For bell-shaped distributions, 95% of all z-scores fall between -2 and +2.
The main use of z-scores is comparison across different scales. You cannot compare a batting average of .420 to a height of 67 inches directly, but you can compare their z-scores.
Worked example: heights of young women are approximately normal with μ = 65 inches and σ = 2 inches.
A woman 67 inches tall: z = (67 - 65) / 2 = 1.0, so she is 1 standard deviation above the mean.
A woman 62 inches tall: z = (62 - 65) / 2 = -1.5, so she is 1.5 standard deviations below the mean.
Worked example (cross-era comparison): Ty Cobb batted .420 in the 1910s (mean .266, SD .0371), Ted Williams batted .406 in the 1940s (mean .267, SD .0326), George Brett batted .390 in the 1970s (mean .261, SD .0317). Their z-scores:
Cobb: z = (.420 - .266) / .0371 ≈ 4.15
Williams: z = (.406 - .267) / .0326 ≈ 4.26
Brett: z = (.390 - .261) / .0317 ≈ 4.07
Williams stands furthest above his peers despite having a lower raw average than Cobb. Z-scores reveal that.
The Pth percentile is the value below which P% of the data falls (when arranged in ascending order), with (100 - P)% above it.
Three special percentiles have their own names:
25th percentile = Q1 (first quartile, lower quartile)
50th percentile = Q2 (second quartile) = the median
75th percentile = Q3 (third quartile, upper quartile)
These three quartiles divide the ordered data into four groups, each containing roughly 25% of the observations.
Arrange the data from smallest to largest.
Find the median. This divides the data into a lower half and an upper half.
Q1 = the median of the lower half (values below the overall median).
Q3 = the median of the upper half (values above the overall median).
Worked example: data set {2, 3, 2, 6, 4, 1, 5, 6, 1}
Sorted: 1, 1, 2, 2, 3, 4, 5, 6, 6
Median = 3 (the 5th value in 9 observations)
Lower half: {1, 1, 2, 2}. Median of this subset = (1 + 2) / 2 = 1.5, so Q1 = 1.5
Upper half: {4, 5, 6, 6}. Median of this subset = (5 + 6) / 2 = 5.5, so Q3 = 5.5
Note: there are several valid methods for computing quartiles and they can give slightly different answers. This course uses the shortcut method above.
IQR = Q3 - Q1
From the worked example above: IQR = 5.5 - 1.5 = 4.0
The IQR captures the spread of the middle 50% of the data. Because it ignores the extremes, it is resistant to outliers. This makes it a better measure of spread than the range when the data contains extreme values.
The range = Maximum - Minimum. It is the simplest measure of spread but is sensitive to outliers; a single extreme value widens it substantially.
Important caveat: the IQR can be misleading when data is heavily concentrated around the median. For example, data set A {10, 10, 10, 10, 10, 10} and data set E {0, 10, 10, 10, 10, 20} both produce an IQR of 0, yet E clearly has more variability.
The five-number summary consists of:
Min (smallest value)
Q1 (first quartile)
Median (m)
Q3 (third quartile)
Max (largest value)
For the worked example above: Min = 1, Q1 = 1.5, Median = 3, Q3 = 5.5, Max = 6.
This summary is the basis for boxplots, which you will encounter shortly. It tells you the full extent of the data (Min to Max), where the middle 50% sits (Q1 to Q3), and where the centre is (Median), all in five numbers.
z = \frac{x - \bar{x}}{s}Z-score for a sample: the data value minus the sample mean, divided by the sample standard deviation.
z = \frac{x - \mu}{\sigma}Z-score for a population: the data value minus the population mean, divided by the population standard deviation.
\text{IQR} = Q_3 - Q_1Interquartile range: the third quartile minus the first quartile.
\text{Range} = \text{Maximum} - \text{Minimum}Range: the largest value minus the smallest value.
Standardised test scores (SAT, GRE) are reported as percentiles precisely because raw scores vary from one test administration to the next. Telling someone they scored at the 85th percentile is more meaningful than giving them a raw number without context. In medicine, growth charts for children use percentiles to flag whether a child's height or weight falls outside the normal range for their age.
Students often confuse percentile with percentage score. Scoring at the 90th percentile does not mean you got 90% of questions right. It means you scored higher than 90% of the people who took the test.
Students sometimes forget that z-scores are unitless. If the original data is in inches, the z-score has no units at all.
When finding quartiles, students frequently include the median in both the lower and upper halves when the data set has an odd number of values. The shortcut method taught in this course excludes the median from both halves.
Students sometimes assume IQR and range measure the same thing. The range spans the entire data set (Min to Max), while the IQR spans only the middle 50% (Q1 to Q3). They can give very different pictures of variability, especially when outliers are present.
⚠️ Z-score computation is a staple exam question. Be able to calculate z from given values and interpret it in a sentence: "The value is [z] standard deviations [above/below] the mean."
⚠️ Cross-dataset comparison via z-scores is a common application question. Expect a scenario where two measurements on different scales need to be compared (like the baseball batting averages example).
⚠️ Quartile and IQR computation from a small data set is frequently tested. Practise the shortcut method until sorting, splitting, and finding sub-medians is automatic.
⚠️ The five-number summary often appears alongside boxplot questions. Know which number corresponds to which feature of the boxplot.
True or false: A z-score of -2.5 means the data value is 2.5 standard deviations below the mean.
True. The negative sign indicates below the mean; the magnitude gives the distance in standard deviations.
Fill in the blank: The IQR measures the spread of the middle ___% of the data.
50%.
True or false: The 50th percentile is another name for the mean.
False. The 50th percentile is the median, not the mean. They are the same only in perfectly symmetric distributions.
True or false: If Q1 = 20 and Q3 = 45, the IQR is 25.
True. IQR = Q3 - Q1 = 45 - 20 = 25.
Fill in the blank: The five-number summary consists of Min, ___, Median, ___, Max.
Q1 and Q3.
Q: Heights of young women are approximately normal with μ = 65 inches and σ = 2 inches. What is the z-score of a woman who is 70 inches tall, and what does it mean?
A: z = (70 - 65) / 2 = 2.5. She is 2.5 standard deviations above the mean height.
Q: For the data set {1, 1, 2, 2, 3, 4, 5, 6, 6}, find Q1, Q3, and the IQR.
A: Median = 3. Lower half = {1, 1, 2, 2}, so Q1 = (1 + 2) / 2 = 1.5. Upper half = {4, 5, 6, 6}, so Q3 = (5 + 6) / 2 = 5.5. IQR = 5.5 - 1.5 = 4.0.
Q: Give the five-number summary for the same data set.
A: Min = 1, Q1 = 1.5, Median = 3, Q3 = 5.5, Max = 6.
Q: Ty Cobb's z-score for batting average in the 1910s is approximately 4.15, while Ted Williams's z-score in the 1940s is approximately 4.26. Who was more dominant relative to their era?
A: Williams. His higher z-score means he stood further above his era's average than Cobb did above his, even though Cobb's raw batting average (.420) was higher.
Q: Data set A = {10, 10, 10, 10, 10, 10} and data set E = {0, 10, 10, 10, 10, 20}. Both have IQR = 0. Does this mean they have the same variability?
A: No. Data set E has values as low as 0 and as high as 20, giving it a range of 20, while data set A has a range of 0. The IQR is misleading here because both sets are concentrated around the median. The standard deviation or range would better distinguish them.
Z-scores are the bridge to the normal distribution, which you will study in depth later. When you learn to find probabilities under the normal curve using z-tables, you will be using exactly this formula. The five-number summary leads directly into boxplots, a graphical tool for comparing distributions side by side. The IQR also plays a role in the 1.5 × IQR rule for identifying outliers, which is covered alongside boxplots.
z-score, standardised value, standardized value, percentile, quartile, Q1, Q2, Q3, first quartile, third quartile, lower quartile, upper quartile, interquartile range, IQR, five-number summary, 5-number summary, range, boxplot, outlier detection, 1.5 IQR rule, normal distribution, standard score, STA 270, introductory statistics, descriptive statistics, Section 2.3