Measuring spread
OpenStax's day class has an interquartile range of 26.5 points, while the evening class has only 11.
A teacher marks the same test in two small classes. The scores are made up. Class A gets 70, 75, 80, 85 and 90. Class B gets 60, 60, 80, 100 and 100. Both means are 80, and a report that gives only the mean would call the classes identical. They are not. In class A every score is within 10 points of 80. In class B four of the five students are 20 points away, two above and two below. The teacher needs a second number, one that says how scattered the scores are. This lesson covers three measures of spread: the range, the interquartile range and the standard deviation. OpenStax's day and evening classes, further down, show the same thing with real data.
The range uses two values; quartiles use the middle half
The range is the largest value minus the smallest. For the two made-up classes it is 90 − 70 = 20 and 100 − 60 = 40. That is easy enough. But it depends on only two values, which are often the least typical ones. Quartiles do better. Sort the data and find the median, also called Q2. The first quartile Q1 is the median of the lower half and the third quartile Q3 is the median of the upper half; when n is odd, OpenStax leaves the median itself out of both halves. About a quarter of the data lies at or below Q1 and about three quarters at or below Q3. The interquartile range, IQR = Q3 − Q1, is the width of the middle half of the data.
OpenStax compares two real statistics classes. The day class has a five-number summary of minimum 32, Q1 56, median 74.5, Q3 82.5 and maximum 99, so its IQR is 82.5 − 56 = 26.5. The evening class has 25.5, 78, 81, 89 and 98, an IQR of 89 − 78 = 11. Both ranges are large, 67 and 72.5. The IQRs tell a different story: the middle half of the evening class is packed far more tightly. A box plot draws these five numbers. The box runs from Q1 to Q3 with a line at the median, and whiskers reach out to the smallest and largest values.
Why the deviations are squared before they are averaged
The standard deviation uses every value. Begin with the deviations from the mean. Adding them up is useless, since they always total 0. So square them first, which makes every one positive, then average the squares and take the square root to get back to the original units. For a sample, OpenStax divides the sum of squares by n − 1 instead of n; the result is the sample standard deviation s. For the quiz scores 6, 8, 9, 7 and 10 (mean 8) the deviations are −2, 0, 1, −1, 2. Their squares add to 10, the sample variance is 10 ÷ 4 = 2.5, and s = √2.5, about 1.58 points. Dividing by n instead gives the population value σ = √2, about 1.41. For the two made-up classes, s is about 7.91 points for class A and exactly 20 for class B.
A rough way to read s: for bell-shaped data, about two thirds of the values lie within one standard deviation of the mean and about 95% within two. Add the same number to every value and neither the IQR nor s changes, because the values move together. Double every value and both double. Like the mean, s is not resistant. One extreme value can inflate it a lot, while the IQR barely notices.
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| Range | Maximum − minimum |
| Q1 and Q3 | Medians of the lower and upper halves |
| IQR | Q3 − Q1, the width of the middle half |
| Five-number summary | Min, Q1, median, Q3, max |
| Sample standard deviation s | √(Σ(x − x̄)² ÷ (n − 1)) |
- What is the range of 2, 4, 4, 4, 5, 5, 7, 9?
- For 2, 4, 4, 4, 5, 5, 7, 9, what is Q1?
- For 2, 4, 4, 4, 5, 5, 7, 9, what is Q3?
- IQR of the OpenStax day class (Q1 = 56, Q3 = 82.5)?
- IQR of the evening class (Q1 = 78, Q3 = 89)?
- Quiz scores 6, 8, 9, 7, 10. Sample variance (sum of squares ÷ (n − 1))?
- Sample standard deviation s of those scores, to 2 decimal places?
- Every score is doubled to 12, 16, 18, 14, 20. What is s now, to 2 decimal places?
- Every score gets 5 bonus points. What happens to the standard deviation?
- Goes up by 5
- Stays the same
- Doubles
- Goes down by 5
- Which measure of spread is least affected by one extreme value?
- Range
- Standard deviation
- IQR
- All are the same
- Calling two sets equally spread because their ranges match: 1, 5, 5, 5, 9 and 1, 1, 5, 9, 9 both have range 8. The range sees only the two ends. In the first set three values sit on the mean, 5; in the second, four values sit at the ends. The IQRs are 4 and 8, and s is about 2.83 against exactly 4.
- Adding the deviations to measure spread: for 2, 4, 6 they are −2, 0 and 2, so "the spread is 0". Deviations from the mean add to 0 for any data. That is why they are squared first. The squares 4, 0 and 4 add to 8.
- Stopping at the variance: for 2, 4, 6, reporting 8 ÷ 2 = 4 as the standard deviation. 4 is the sample variance, measured in squared units. Take the square root: s = 2, in the same units as the data.
- Range = max − min; simple but fragile.
- Q1, Q3 = medians of the halves; IQR = Q3 − Q1 is resistant.
- Sample standard deviation s divides the squared deviations by n − 1.
- Shifting data leaves spread unchanged; scaling scales it.
Sources for this course (7)
- Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 5~6학년군 [6수04-01] 평균의 의미, 고려 사항 「평균을 구하는 방법뿐만 아니라 그 의미를 직관적으로 파악하게 한다」; 중학교 1~3학년군 [9수04-01] 중앙값·최빈값과 대푯값 고르기, [9수04-07] 분산과 표준편차, [9수04-08] 상자그림으로 분포 비교, 용어 변량·대푯값·중앙값·최빈값·도수분포표·산포도·편차·분산·표준편차·사분위수·상자그림; 고등학교 확률과 통계 용어 표본평균·모평균·표본분산·표본표준편차와 기호 X̄, S², S, N(m, σ²); 실용 통계 [12실통02-04] 대푯값과 산포도 고르기)
- Middle-school variance as the mean of the squared deviations: EBS Math 중3 「분산과 표준편차」 (www.ebsmath.co.kr/resource/rscView?cate=123500&cate2=123510&cate3=123511&rscTpDscd=RTP10&grdCd=MGRD03&sno=24525&type=S, read 2026-10-19)
- Textbook: OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean, 2.3 measures of location, quartiles as medians of the lower and upper halves, IQR, 1.5 × IQR outlier rule, 13 real estate prices, 11 salaries, day and evening class test scores (openstax.org/books/introductory-statistics-2e/pages/2-3-measures-of-the-location-of-the-data)
- 2.5 measures of the center (.../pages/2-5-measures-of-the-center-of-the-data)
- 2.6 skewness and the mean, median and mode, the three ten- and sixteen-value data sets (.../pages/2-6-skewness-and-the-mean-median-and-mode)
- 2.7 measures of the spread, sample standard deviation with n − 1 (.../pages/2-7-measures-of-the-spread-of-the-data)
- Trimmed mean only: Federal Reserve Bank of Dallas, Trimmed Mean PCE inflation rate (dallasfed.org/research/pce and /research/pce/descr)
Checked: 2026-10-17