Difficulty: Introductory | Prerequisites: Section 2.2 notes (measures of centre: mean, median, mode). You should be comfortable calculating a mean before starting here.
This topic sits in the early descriptive-statistics block of the course. You have already learned how to find the centre of a data set (mean, median, mode). Now the question becomes: how spread out are the values around that centre? Two data sets can share the same mean and median yet behave very differently, and measures of spread are what reveal that difference. Without them, you are describing a distribution with one number and missing half the story.
If you skipped the section on measures of centre, go back and cover it first. Everything here builds on being able to compute a mean.
Measures of spread tell you how far data values sit from the centre. The three main ones are the standard deviation (typical distance from the mean), the interquartile range (spread of the middle 50%), and the range (gap between the largest and smallest values). For bell-shaped distributions, the empirical rule lets you estimate what proportion of data falls within one, two, or three standard deviations of the mean.
Variability (spread)
How much the data values differ from one another and from the centre. In simple terms, this is the "width" of your data.
Deviation from the mean
The difference between an individual data value and the mean, written as x_i - x̄. Think of it as how far one observation sits from the average.
Sample variance (s²)
The average of the squared deviations from the mean, divided by n - 1 (not n). It measures spread in squared units. In simple terms, it is the "average squared distance" each point sits from the mean, with a small correction for using a sample.
Sample standard deviation (s)
The square root of the sample variance. It brings the measure of spread back into the same units as the original data. Think of it as the typical distance a data value falls from the mean.
Population standard deviation (σ)
The standard deviation calculated using every value in the entire population, rather than a sample. The formula divides by N instead of n - 1.
Empirical rule (68-95-99.7 rule)
A guideline for bell-shaped distributions: roughly 68% of values fall within one standard deviation of the mean, 95% within two, and nearly all within three. In simple terms, if your histogram looks like a bell curve, these three percentages give you a quick feel for where the data sits.
Range
The simplest measure of spread: Maximum - Minimum. Quick to compute but easily distorted by a single extreme value.
Resistance to outliers
A measure is "resistant" if extreme values do not pull it around much. The median and IQR are resistant; the mean, standard deviation, and range are not.
Whenever you describe a quantitative variable, cover three things:
Centre (where the data clusters): mean, median, mode
Spread (how far values range from that centre): standard deviation, IQR, range
Shape (the overall pattern): symmetric, bell-shaped, skewed left or right
Centre alone is not enough. Five data sets can share the exact same mean and median yet differ wildly in how tightly or loosely values cluster around that centre. That is why spread is the second feature you report.
Consider five data sets, each with x̄ = 10 and median = 10:
Data set A: 10, 10, 10, 10, 10, 10 (no spread at all)
Data set D: 7, 8, 9, 11, 12, 13 (moderate spread)
Data set B: 5, 5, 5, 15, 15, 15 (wider spread)
Data set E: 0, 10, 10, 10, 10, 20 (concentrated at the centre with extreme tails)
Data set C: 0, 0, 0, 20, 20, 20 (maximum spread)
All five share the same centre, but they tell very different stories. A measure of spread captures that difference numerically.
The standard deviation quantifies the typical distance of data values from the mean.
Steps to compute the sample standard deviation:
Calculate the mean x̄.
Subtract the mean from each data value to get the deviations (x_i - x̄).
Square each deviation.
Sum the squared deviations.
Divide that sum by n - 1 (this gives the sample variance s²).
Take the square root of the variance (this gives s).
Worked example with Data set D (7, 8, 9, 11, 12, 13), where x̄ = 10:
Deviations: -3, -2, -1, 1, 2, 3
Squared deviations: 9, 4, 1, 1, 4, 9
Sum of squared deviations: 28
Variance: s² = 28 / (6 - 1) = 28 / 5 = 5.6
Standard deviation: s = √5.6 ≈ 2.366
Interpretation: the values in Data set D typically fall about 2.37 units away from the mean of 10.
s is always greater than or equal to zero. It equals zero only when every data value is the same (no variability at all).
Greater variability in the data produces a larger s.
s is in the same units as the data (e.g. inches, bpm, hours). The variance s² is in squared units, which is harder to interpret directly.
s is not resistant to outliers. A single extreme value can inflate it substantially.
This rule applies only to distributions that are roughly bell-shaped (mound-shaped, symmetric, one peak).
About 68% of all values fall within one standard deviation of the mean: the interval [x̄ - s, x̄ + s].
About 95% fall within two standard deviations: [x̄ - 2s, x̄ + 2s].
Nearly all values (about 99.7%) fall within three standard deviations: [x̄ - 3s, x̄ + 3s].
Worked example: pulse rates for college students are approximately bell-shaped with mean 69.6 bpm and standard deviation 12.2 bpm. The interval containing about 95% of values is:
69.6 ± 2(12.2) = 69.6 ± 24.4, which gives 45.2 to 94.0 bpm.
The empirical rule is a quick estimation tool. If someone asks "within what range do most values fall?", two standard deviations either side of the mean covers roughly 95% of them.
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}Sample variance: sum the squared deviations from the mean, then divide by n - 1.
s = \sqrt{s^2}Sample standard deviation: the square root of the variance.
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}Population standard deviation: uses the population mean μ and divides by the total population size N (not N - 1).
Empirical rule intervals (bell-shaped distributions only):
68% of data: [x̄ - s, x̄ + s]
95% of data: [x̄ - 2s, x̄ + 2s]
99.7% of data: [x̄ - 3s, x̄ + 3s]
Manufacturing uses standard deviation to monitor quality: if the spread of part dimensions exceeds a threshold, the process is out of control and needs adjustment. In finance, the standard deviation of returns is a standard measure of investment risk, so when someone says a fund is "high volatility", they mean its returns have a large standard deviation.
Students often divide by n instead of n - 1 when computing sample variance. The n - 1 denominator (called Bessel's correction) is used for samples; dividing by N is only for the full population.
Students sometimes think a standard deviation of zero means missing data. It does not. It means every value in the data set is identical.
Students sometimes confuse the units: the variance s² is in squared units (e.g. square inches), while the standard deviation s is in the original units (inches). Exam answers that report the wrong unit lose marks.
The empirical rule is frequently misapplied to skewed distributions. It works only when the histogram is roughly bell-shaped. If the distribution is heavily skewed, the 68-95-99.7 percentages do not hold.
⚠️ You will almost certainly be asked to compute a standard deviation by hand from a small data set. Practise the six-step process until it is automatic.
⚠️ Empirical rule questions are common: given a mean and standard deviation, find the interval that contains 68% or 95% of the data, or work backwards to find what proportion lies beyond a given value.
⚠️ Know which measures are resistant to outliers (median, IQR) and which are not (mean, standard deviation, range). This comparison appears regularly on exams.
⚠️ Be ready to interpret s in context: "the data values typically fall about [s] [units] from the mean."
True or false: The standard deviation can be negative.
False. Because s² is a sum of squared values divided by a positive number, s² ≥ 0, and its square root s ≥ 0.
True or false: If every value in a data set is the same, the standard deviation is 0.
True. Every deviation from the mean is 0, so the sum of squared deviations is 0.
Fill in the blank: For a bell-shaped distribution, about ___% of data falls within two standard deviations of the mean.
95%.
True or false: The variance and the standard deviation are measured in the same units.
False. The variance is in squared units; the standard deviation is in the original units.
True or false: The empirical rule works well for any distribution shape.
False. It applies only to distributions that are approximately bell-shaped (symmetric, single peak).
Q: The average clean-up time for a crew is 84.0 hours with a standard deviation of 6.8 hours. Assuming the empirical rule applies, within what interval will the clean-up time fall 95% of the time?
A: 84.0 ± 2(6.8) = 84.0 ± 13.6, so the interval is 70.4 to 97.6 hours.
Q: Using the same clean-up data (mean 84.0, s = 6.8), what proportion of the time will it take the crew 97.6 or more hours?
A: 97.6 is exactly two standard deviations above the mean. The empirical rule says 95% of values fall within two standard deviations, leaving 5% in the tails combined. Because the distribution is symmetric, half of that 5% sits in the upper tail: about 2.5%.
Q: Data set A is {10, 10, 10, 10, 10, 10}. What is its standard deviation?
A: s = 0. Every value equals the mean, so every deviation is 0.
Q: Why do we divide by n - 1 rather than n when computing the sample variance?
A: Dividing by n - 1 (Bessel's correction) gives an unbiased estimate of the population variance. Dividing by n would systematically underestimate it.
Q: A histogram is strongly right-skewed. A classmate applies the empirical rule to estimate the proportion of data within one standard deviation of the mean. Is this appropriate?
A: No. The empirical rule requires a roughly bell-shaped (symmetric, single-peak) distribution. Applying it to a skewed distribution will give misleading results.
This material connects directly to z-scores and percentiles (covered in Part 2 of these notes), which use the standard deviation to express how far a single observation sits from the mean on a standardised scale. It also provides the foundation for boxplots, which visualise the five-number summary and use the IQR to flag outliers. Later in the course, standard deviation reappears in confidence intervals and hypothesis testing, where it helps quantify how much sample statistics vary from sample to sample.
standard deviation, variance, empirical rule, 68-95-99.7 rule, measures of spread, measures of variability, sample variance, population variance, s squared, sigma, deviation from the mean, bell-shaped distribution, mound-shaped, resistance to outliers, spread vs centre, STA 270, introductory statistics, descriptive statistics, Section 2.3