Range and standard deviation.
Center tells us where the data sit. Spread tells us how tightly. Range is the quick read; standard deviation is the precise one. Both set up the bell curve in Lesson 5.
Two classes both averaged 75% on the midterm. From the principal's report — just the mean — the two classes look identical. But in Class A almost everyone landed within a few points of 75; in Class B half the students failed and the other half aced it. Same mean. Wildly different story.
The number that captures that difference is the standard deviation. It measures how spread out the data is around the mean. A quick warm-up version of the same idea is the range, just max − min. Range gives a rough sketch; standard deviation gives the precise reading.
By the end of this lesson, you'll compute the range of any data set in seconds (it's the ALEKS Q4 ask) and understand what standard deviation measures and how Excel computes it. Lesson 5's empirical rule rides directly on standard deviation, so getting comfortable with what SD means now pays off twice.
Center is half the story. Spread is the other half.
Standard deviation (SD) looks at every data point. It is the typical distance from the mean. A tightly clustered data set has a small SD (each point sits close to the mean); a widely spread data set has a large SD (points are typically far from the mean). The formal definition involves squaring deviations and taking a square root at the end, but the intuition is just "the typical gap between a value and the mean."
On exams and in spreadsheets, we let the tool do the arithmetic. Excel has
=STDEV.S() for a sample standard deviation (divide by n − 1) and =STDEV.P() for a population standard deviation (divide by n). For most ALEKS-style problems on data sets, the sample SD is the right one.Both classes have the same mean (75), but Class A's scores cluster tightly (range = 5, SD ≈ 1.8) while Class B's are spread out (range = 28, SD ≈ 13.5). Center alone cannot tell those apart; spread does.
A measure of spread is a single number that summarizes how tightly or loosely a data set is scattered around its center. Where the measures of center (Lesson 2) ask where, measures of spread ask how wide.
Words you'll see on ALEKS and in Excel
- Range Maximum value minus minimum value. Always non-negative. ALEKS Q4 is a direct range calculation: max − min, in whatever units the data carries (pounds, dollars, hours).
- Standard deviation (SD or σ) The typical (root-mean-square) distance from the mean. Has the same units as the data. A data set in pounds has an SD in pounds; a data set in dollars has an SD in dollars.
-
Sample SD vs population SD
Sample SD divides the sum of squared deviations by n − 1; population SD divides by n. Most real data sets are samples, so the sample version is the default. Excel:
=STDEV.S()(sample) vs=STDEV.P()(population). - Variance The SD without the final square root. Variance has units of data² (pounds squared, dollars squared), which is why the SD is usually reported instead — its units match the data.
- Outlier A value far from the rest of the data. Outliers pull the mean and the SD upward (both are sensitive). The median and the range are less affected: the median barely moves, and the range only shifts if the outlier is the new max or min.
Range and SD on the same data set.
"The veterinarian treated 7 dogs this morning. Their weights in pounds: 26, 71, 56, 3, 5, 26, 24. Find the range and the (sample) standard deviation, rounded to one decimal place."
Sort the data (optional but helpful).
Sorting makes the max and min obvious and is harmless. The min is the first sorted value; the max is the last.
Range = max − min.
Read off the two ends of the sorted list and subtract. The units are the data's units: pounds.
Compute the mean (we need it for SD).
SD measures distance from the mean, so we need the mean first. Sum the seven weights, divide by 7.
SD by Excel (or calculator).
Paste the seven values into a column and ask Excel for the sample standard deviation. The formula is =STDEV.S(A1:A7) if the weights live in cells A1 through A7.
Sanity check.
The range (68) is much larger than the SD (24.9), as it always should be: range captures the extremes, SD captures the typical gap. A rough rule of thumb for moderate sample sizes is SD ≈ range / 4, which here gives 17 — close enough for a sanity check, though the 71-pound outlier inflates our SD past that estimate.
Three problems. Range is the ALEKS target.
Find the range of: 4, 12, 7, 18, 9, 15, 6.
Seven dogs at the vet weigh, in pounds: 26, 71, 56, 3, 5, 26, 24. Find the range.
Both data sets below have a mean of 50. Set A: 48, 49, 50, 51, 52. Set B: 20, 30, 50, 70, 80. Which set has the larger standard deviation? Type A or B.
Three fast questions before you move on.
Q1. What is the range of a data set?
Q2. Two data sets have identical means. Set X has values tightly clustered near the mean; Set Y has values far from the mean. Which statement is true?
Q3. Which Excel function computes the sample standard deviation of values in cells A1 through A20?
=STDEV.S is the sample standard deviation (divide by n − 1). =STDEV.P is the population SD; =VAR.S is the sample variance (SD squared); =AVERAGE is the mean. The .S suffix is the giveaway for sample versions.
From numbers to pictures.
Center (Lesson 2) and spread (Lesson 3) together describe a data set with two numbers. The fastest way to communicate both at once is a picture. Lesson 4 introduces the four data displays that make most of the journalistic charts you see — histograms, bar charts, box plots, and pie charts — and shows which one to match to which question.
Next: Lesson 4 covers reading data displays. Lesson 5 then comes back to standard deviation in a starring role: the empirical rule (68-95-99.7) measures exactly how data spreads around the mean inside a bell curve, and it's the source of ALEKS review Q5.
Continue to Lesson 04Different angle? Need another rep? These are optional — tap any that look helpful.
How To Calculate The Standard Deviation
Walks through s = √(Σ(x − x̄)² / (n − 1)) step by step with two examples, explicitly explaining why we divide by n − 1. Exactly the formula in this lesson.
Measures of Spread — Crash Course Statistics #4
Conceptual framing of range, variance, and SD as 'typical distance from the mean.' Pairs well with OCT's procedural treatment above.