math.wiki
Statistics › Center and spread › Standard deviation › Interpreting a standard deviation
Open on the map
Algebra 1 · S.ID.A.2 · S.ID.A.3

Interpreting a standard deviation

A machine fills juice bottles to 500 mL on average. How far from 500 mL should any one bottle be expected to land?

You need
Resistant and non resistant measuresMath 6Individuals, variables and unitsMath 6
Where you are 3 in Standard deviation
Statistics7
Center and spread13
Standard deviation3
Computing a standard deviation, sample and populationVariance of a data set
Interpreting a standard deviation

When you reach for it

A standard deviation arrives as one number with a unit, such as 2 mL. You reach for this skill when that number must become a statement about the data: what a typical value looks like, which group is more spread out, whether one value is far out, and whether the standard deviation is the right summary at all.

The procedure

What the number measures. The standard deviation is the square root of the variance, the mean of the squared distances from the Mean \mu over all N values:

\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}

Squaring and then taking the root brings the answer back to the units of the data, so the standard deviation of volumes in milliliters is in milliliters. It is roughly the distance a typical value sits from the mean. The working itself, and the sample version s, which divides by n - 1 and comes out a little larger, belong to Computing a standard deviation, sample and population. Every data set here is a whole group (one batch, one week), so it divides by N; each interpretation reads the same for s.

Typical, not exact, not a maximum. Being the root of a mean of squares, the standard deviation blends all the distances. Often no value sits exactly one standard deviation from the mean, and some values sit further out. It is never smaller than the mean absolute deviation (MAD), because squaring gives the larger distances more weight, and the two are equal only when every value sits the same distance from the mean.

The sentence. An interpretation names the individuals, the variable, its units and the mean, so that someone who has not seen the data can follow it (Individuals, variables and units). A sentence of this shape does it: the bottles in Machine A’s batch typically hold about 2 mL more or less than their mean of 500 mL. The word “about” is honest, because the standard deviation is a rough typical distance.

Comparing two groups. The larger standard deviation belongs to the group whose values stray further from their own mean. It says nothing about where either group is centered; that is the means’ job. So a comparison states both: which center is higher, and which group is more spread. (Comparing two populations: the gap between means in MADs measured the gap between two means in MADs; here the spread itself is the subject.)

Reading it from a picture. On a dot plot or histogram, look at where the bulk of the values sits relative to the mean, not only at the two ends. Values piled near the mean give a small standard deviation; values piled far out on both sides give a large one, even with the same range. A Relative frequency histograms is read the same way, with bar heights as shares of the data.

Counting standard deviations. To judge whether one value is far out, divide its distance from the mean by the standard deviation. The count has no unit, which is what lets a value in one data set be compared with a value in another.

One far value. Every value enters through a square, so a value far from the mean adds a huge square and can pull the standard deviation up a long way. The interquartile range (IQR) depends only on the quartiles, which sit in the middle of each half, so moving one end value barely changes it, often not at all. The standard deviation is not resistant; the IQR is. That decides which pair to report: mean with standard deviation for data that are roughly symmetric with no far values, median with IQR for skewed data or data with outliers.

Worked examples

Example 1

What 2 mL means

All data in this lesson are made up. A juice plant’s Machine A fills a batch of ten bottles with these volumes, in milliliters: 497, 497, 499, 499, 500, 500, 501, 501, 503, 503. Check that the mean is 500 mL and the standard deviation 2 mL, interpret the 2 mL in one sentence, and say how many bottles sit exactly 2 mL from the mean.

Solution

The values add to 5000, so the mean is \displaystyle \frac{5000}{10} = 500 mL. The distances from the mean are 3, 3, 1, 1, 0, 0, 1, 1, 3, 3, and their squares 9, 9, 1, 1, 0, 0, 1, 1, 9, 9 add to 40. The variance is \displaystyle \frac{40}{10} = 4 \text{mL}^2, and the standard deviation is \displaystyle \sqrt{4} = 2 mL.

The bottles in Machine A’s batch typically hold about 2 mL more or less than their mean of 500 mL.

No bottle sits exactly 2 mL away: the distances are 0, 1 and 3. The 2 mL blends them, and four bottles sit further out, at 3 mL. The standard deviation is neither the distance of any particular bottle nor the largest distance.

Example 2

Same mean, different spread

Machine B fills a batch of ten bottles: 491, 492, 495, 497, 499, 501, 503, 505, 508, 509 mL. Its mean is 500 mL and its standard deviation 6 mL. Compare the two machines.

Solution

A quick check of B: the distances from 500 are 9, 8, 5, 3, 1, 1, 3, 5, 8, 9 and their squares add up as 81 + 64 + 25 + 9 + 1 + 1 + 9 + 25 + 64 + 81 = 360 so the variance is \displaystyle \frac{360}{10} = 36 and the standard deviation \displaystyle \sqrt{36} = 6 mL.

Machine A: mean 500 mL, SD 2 mL 486 490 494 498 502 506 510 514 Machine B: mean 500 mL, SD 6 mL 486 490 494 498 502 506 510 514 fill volume (mL)
Figure 1Both machines are centered on 500 mL. The shaded band runs from one standard deviation below the mean to one above, and Machine B’s is three times as wide.

Both machines are centered on 500 mL, so neither overfills or underfills on average. They differ in consistency: a Machine B bottle typically sits about 6 mL from 500 mL, three times Machine A’s 2 mL. For bottles close to the label, A is the better machine.

Example 3

Same spread, different means

Seven made-up July daily highs, in °C, for two inland towns: the valley town 27, 27, 30, 31, 31, 32, 32 and the plateau town 18, 21, 22, 22, 23, 23, 25. Find each mean and standard deviation and compare the towns.

Solution

Valley: the sum is 210, so the mean is \displaystyle \frac{210}{7} = 30 °C. The distances are 3, 3, 0, 1, 1, 2, 2, with squares adding to 9 + 9 + 0 + 1 + 1 + 4 + 4 = 28, so the variance is \displaystyle \frac{28}{7} = 4 and the standard deviation 2 °C.

Plateau: the sum is 154, so the mean is \displaystyle \frac{154}{7} = 22 °C. The distances are 4, 1, 0, 0, 1, 1, 3, with squares adding to 16 + 1 + 0 + 0 + 1 + 1 + 9 = 28, so again the variance is 4 and the standard deviation 2 °C.

Valley town: mean 30 °C, SD 2 °C 16 18 20 22 24 26 28 30 32 34 Plateau town: mean 22 °C, SD 2 °C 16 18 20 22 24 26 28 30 32 34 daily high (°C)
Figure 2Different centers, the same spread. The bands are equally wide; they sit 8 °C apart.

The valley town was warmer, its highs centered on 30 °C against 22 °C, a gap of 8 °C. The day-to-day variation was the same: in each town a day’s high typically sat about 2 °C from that town’s own mean. Neither number answers the other’s question, so a full comparison gives both.

Example 4

Matching standard deviations to pictures

The dot plots show made-up data sets P, Q and R, ten values each. All three have mean 5 and run from 2 to 8. Their standard deviations, rounded, are 1.4, 2.1 and 2.7. Match them without computing.

P 0 1 2 3 4 5 6 7 8 9 10 Q 0 1 2 3 4 5 6 7 8 9 10 R 0 1 2 3 4 5 6 7 8 9 10 value
Figure 3Same mean, same range. Where the values in between sit decides the standard deviation.
Solution

The range is 6 in each, so the ends cannot tell the plots apart; what matters is where the values in between sit. P has six of its ten values at the mean itself and only two at the ends, so its typical distance is small: 1.4. R has eight values at the ends, 3 from the mean, and two at the mean, so its typical distance is close to 3: 2.7. Q spreads its values across the whole range and falls between: 2.1.

A check on R: the squared distances are eight 9s and two 0s, adding to 72, so the variance is 7.2 and \displaystyle \sqrt{7.2} \approx 2.68, which rounds to 2.7.

Example 5

How far is far

Machine B (mean 500 mL, standard deviation 6 mL) fills a bottle with 488 mL. How many standard deviations below the mean is it? How would the same bottle look had it come from Machine A (mean 500 mL, standard deviation 2 mL)?

Solution

The bottle is 500 - 488 = 12 mL short. For Machine B that is \displaystyle \frac{12}{6} = 2 standard deviations below the mean: further out than any bottle in Example 2, whose furthest sat \displaystyle \frac{9}{6} = 1.5 standard deviations away, but not wildly so.

For Machine A the same 12 mL is \displaystyle \frac{12}{2} = 6 standard deviations below the mean. A’s furthest bottle in Example 1 sat \displaystyle \frac{3}{2} = 1.5 standard deviations away, so a 488 mL bottle from A would point to a fault. The same distance in milliliters is ordinary for one machine and extreme for the other, and counting in standard deviations is what shows it. This count, with a sign for above or below, becomes the z-score in a later lesson.

Example 6

One jammed bottle

A nozzle jams on one of Machine A’s bottles, so the batch from Example 1 becomes 467, 497, 499, 499, 500, 500, 501, 501, 503, 503 mL: one 497 mL bottle is replaced by one holding 467 mL. Find the mean, standard deviation, median and IQR, compare them with the original batch, and decide which pair describes this batch.

Solution

The sum falls by 30 to 4970, so the mean is \displaystyle \frac{4970}{10} = 497 mL. The distances from 497 are 30, 0, 2, 2, 3, 3, 4, 4, 6, 6 and their squares add up as 900 + 0 + 4 + 4 + 9 + 9 + 16 + 16 + 36 + 36 = 1030 so the variance is \displaystyle \frac{1030}{10} = 103 and the standard deviation \displaystyle \sqrt{103} \approx 10.1 mL.

The 5th and 6th values are both 500, so the median is 500 mL, as before. The lower half 467, 497, 499, 499, 500 has middle value Q_1 = 499 and the upper half 500, 501, 501, 503, 503 has Q_3 = 501, so the IQR is 2 mL. The original lower half 497, 497, 499, 499, 500 had the same middle value, so the IQR was 2 mL then too.

Machine A batch with one jammed bottle 464 468 472 476 480 484 488 492 496 500 504 508 fill volume (mL) mean ± SD IQR (middle half)
Figure 4One jammed bottle stretches the standard deviation to about 10.1 mL. The IQR stays 2 mL.
in mLbeforejammed
mean500497
standard deviation210.1
median500500
IQR22

One bottle multiplied the standard deviation by about five, because its single square, 900, is most of the 1030; the IQR did not move. Read literally, “a typical bottle sits about 10.1 mL from 497 mL” describes no bottle at all: nine sit within 3 mL of 500 mL and one sits 33 mL below. So this batch is reported as median 500 mL with IQR 2 mL, and the jammed bottle is noted on its own. The original batch, symmetric with no far value, was well described by its mean and standard deviation.

Boundaries

  • A standard deviation needs numerical data with a meaningful mean. Shirt sizes coded 1 to 4 have a standard deviation that says nothing.
  • It describes spread around the mean, not shape. Two data sets can share a mean and a standard deviation and still look quite different, so the plot belongs beside the number.
  • “Typical distance” is rough. The standard deviation is at least the MAD, equal to it only when every distance is the same (Exercise 4), and it is 0 only when every value is the same.
  • It is not resistant: for skewed data or data with outliers the median with the IQR is the better summary.
  • A count of standard deviations says how unusual a value is only relative to one data set, and only when the standard deviation suits that data.

Where this goes next

Counting standard deviations becomes the z-score, and for bell-shaped data the normal distribution later says what share of the values lies within one or two standard deviations of the mean. Adding the same amount to every value, or multiplying every value by a constant, moves the mean and the standard deviation in predictable ways: Shifting and rescaling a data set.

Exercises

Saying what a standard deviation means

  1. Machine C at the juice plant fills one-liter cartons. Its fills have mean 1004 mL and standard deviation 3 mL. Write one sentence interpreting the 3 mL.
  2. A reader of the summary in Exercise 1 concludes, “So a typical carton holds 3 mL.” Explain what is mixed up, and say roughly what a typical carton does hold.
  3. Andrzej records the plateau town’s daily highs for five days in August: 20, 22, 23, 24, 26 °C. Find the mean and standard deviation, and interpret the standard deviation in one sentence.
  4. Machine D fills eight small bottles: 249, 249, 249, 249, 251, 251, 251, 251 mL. Find the standard deviation. Why is it here exactly the distance of every bottle from the mean?

Comparing spreads

  1. Two other inland towns both have a mean July high of 26 °C. Town East’s standard deviation is 1.5 °C and Town West’s is 4.5 °C. Which town’s July weather is steadier, and how many times as spread out is the other?
  2. Machine A has mean 500 mL and standard deviation 2 mL; Machine E has mean 506 mL and standard deviation 2 mL. Compare the two machines in two sentences.
  3. Without computing, order these from smallest standard deviation to largest, then compute each to check. J: 10, 10, 10, 10, 10; K: 8, 9, 10, 11, 12; L: 4, 8, 10, 12, 16.
  4. The sets 1, 5, 5, 5, 9 and 1, 1, 5, 9, 9 have the same mean and the same range. Which has the larger standard deviation? Find both to two decimal places.

Counting standard deviations

  1. Machine B has mean 500 mL and standard deviation 6 mL. How many standard deviations from the mean, and on which side, are bottles of 509 mL and 485 mL?
  2. The valley town’s July highs have mean 30 °C and standard deviation 2 °C; the plateau town’s have mean 22 °C and standard deviation 2 °C. One day each town reaches 25 °C. In which town is that day further from usual, counted in standard deviations?
  3. For Machine B, what fill volume is 3 standard deviations above the mean? What volume is 1.5 standard deviations below it?
  4. Apolline says a bottle 4 mL short from Machine A (standard deviation 2 mL) is further out than a bottle 10 mL short from Machine B (standard deviation 6 mL). Is she right?

One far value

  1. The plateau town’s week in Example 3 had mean 22 °C, standard deviation 2 °C, median 22 °C and IQR 2 °C. A faulty sensor records the last day as 39 °C instead of 25 °C. Find the new mean, standard deviation (two decimal places), median and IQR.
  2. Over ten days the plant’s bottling line stopped for these numbers of minutes: 0, 0, 0, 1, 1, 2, 2, 3, 5, 26. Find the mean, standard deviation (two decimal places), median and IQR, and say which pair to report.
  3. In Example 6, suppose the jammed bottle had held 437 mL instead of 467 mL. Find the new mean, standard deviation (two decimal places) and IQR. What do the results show?

Reasoning

  1. Can a standard deviation be negative? When is it 0, and what would a standard deviation of 0 mL say about a batch of bottles?
  2. Explain why pushing the smallest value of a data set much further down can change the standard deviation a great deal but leave the IQR unchanged.
  3. Is the standard deviation the furthest any value sits from the mean? Answer using Machine A’s batch from Example 1, and say what the standard deviation is instead.
Answers
  1. A carton filled by Machine C typically holds about 3 mL more or less than the mean of 1004 mL.
  2. The 3 mL is a distance from the mean, not an amount of juice. A typical carton holds about 1004 mL, give or take about 3 mL.
  3. Mean \displaystyle \frac{115}{5} = 23 °C. Distances 3, 1, 0, 1, 3, squares adding to 20, variance \displaystyle \frac{20}{5} = 4, standard deviation 2 °C. On those five days the plateau town’s high typically sat about 2 °C from the mean of 23 °C.
  4. Mean \displaystyle \frac{2000}{8} = 250 mL. Every bottle is exactly 1 mL from it, so every square is 1, the variance is \displaystyle \frac{8}{8} = 1 and the standard deviation is 1 mL. When all the distances are equal, the mean of their squares is that distance squared, and the root gives it back.
  5. Town East is steadier. Town West is \displaystyle \frac{4.5}{1.5} = 3 times as spread out: its highs typically sit about 4.5 °C from 26 °C, against 1.5 °C in Town East.
  6. Machine E fills 506 - 500 = 6 mL more on average. The two are equally consistent: on each, a bottle typically sits about 2 mL from that machine’s mean.
  7. J, then K, then L. J: every value is the mean, so the standard deviation is 0. K: squares 4, 1, 0, 1, 4, sum 10, variance 2, standard deviation \displaystyle \sqrt{2} \approx 1.41. L: squares 36, 4, 0, 4, 36, sum 80, variance 16, standard deviation 4.
  8. The second. Both means are 5. First: squares 16, 0, 0, 0, 16, variance \displaystyle \frac{32}{5} = 6.4, standard deviation \approx 2.53. Second: squares 16, 16, 0, 16, 16, variance \displaystyle \frac{64}{5} = 12.8, standard deviation \approx 3.58. It has four values at the ends instead of two.
  9. 509 mL is \displaystyle \frac{9}{6} = 1.5 standard deviations above; 485 mL is \displaystyle \frac{15}{6} = 2.5 standard deviations below.
  10. The valley town: there 25 °C is \displaystyle \frac{5}{2} = 2.5 standard deviations below the mean, while in the plateau town it is \displaystyle \frac{3}{2} = 1.5 standard deviations above.
  11. 500 + 3 \times 6 = 518 mL, and 500 - 1.5 \times 6 = 491 mL.
  12. Yes. Machine A: \displaystyle \frac{4}{2} = 2 standard deviations. Machine B: \displaystyle \frac{10}{6} \approx 1.67 standard deviations. Fewer milliliters, but further out for its own machine.
  13. The data become 18, 21, 22, 22, 23, 23, 39. Mean \displaystyle \frac{168}{7} = 24 °C. Distances 6, 3, 2, 2, 1, 1, 15, squares adding to 280, variance 40, standard deviation \displaystyle \sqrt{40} \approx 6.32 °C. The median is still the 4th value, 22 °C. Lower half 18, 21, 22 gives Q_1 = 21, upper half 23, 23, 39 gives Q_3 = 23, so the IQR is still 2 °C. One reading more than tripled the standard deviation and left the median and IQR alone.
  14. Mean \displaystyle \frac{40}{10} = 4 minutes. Distances 4, 4, 4, 3, 3, 2, 2, 1, 1, 22, squares adding to 560, variance 56, standard deviation \displaystyle \sqrt{56} \approx 7.48 minutes. Median \displaystyle \frac{1 + 2}{2} = 1.5 minutes; lower half 0, 0, 0, 1, 1 gives Q_1 = 0, upper half 2, 2, 3, 5, 26 gives Q_3 = 3, so IQR 3 minutes. The data are skewed right with one far value, so report the median and IQR: the mean of 4 is above eight of the ten days, and a standard deviation of 7.48 minutes is larger than every stoppage but one.
  15. Sum 4940, mean \displaystyle \frac{4940}{10} = 494 mL. Distances 57, 3, 5, 5, 6, 6, 7, 7, 9, 9, squares adding to 3640, variance 364, standard deviation \displaystyle \sqrt{364} \approx 19.08 mL. The halves still have middle values Q_1 = 499 and Q_3 = 501, so IQR 2 mL. The further the one bottle falls, the larger the standard deviation grows (2, then 10.1, then 19.08 mL), while the IQR does not notice.
  16. No: it is the square root of a mean of squares, which is never negative. It is 0 exactly when every distance is 0, that is, every value equals the mean. A standard deviation of 0 mL would mean every bottle in the batch held exactly the same volume.
  17. The standard deviation squares every distance from the mean, so a value pushed far down adds a very large square. The IQR depends only on Q_1 and Q_3, the middle values of the two halves. The smallest value stays the smallest and stays in the lower half, and as long as each half has at least three values it is never that half’s middle value, so neither quartile changes.
  18. No. Machine A’s distances from the mean are 3, 3, 1, 1, 0, 0, 1, 1, 3, 3 mL. The furthest is 3 mL, but the standard deviation is 2 mL. It is the square root of the mean of the squared distances: a typical distance that blends the small ones with the large ones.
kg1128 · updated 2026-10-10 · CC-BY-SA-4.0