"Accurate to within plus or minus 3.1 percentage points, 19 times out of 20." The line closes countless poll reports, and most readers skim past it. A few pages later in a medical journal, a table says the treated group scored 103 ± 2.5. A statistics textbook reports that the middle half of a class of students is between 161 and 171 cm tall. Three ranges, one symbol or its cousin, and three entirely different claims about the world.
Mixing them up is not a small slip. Read a standard error as a standard deviation and the data look six times more consistent than they were. Read a poll's margin as the spread of opinion and a 52% to 48% race looks settled when it is not. This article puts four ranges side by side, the standard deviation, the standard error, the margin of error and the quartile range, runs them against each other in three rounds, and ends with a verdict on which one answers which question. Every number comes from the confidence interval calculator, the standard error calculator and the grouped data quartile calculator.
Four ranges side by side
Before the rounds, the four contenders on one page. The row that matters most is the third: two of these ranges shrink as you collect more data, and two never do.
| Standard deviation | Standard error | Margin of error | Quartile range | |
|---|---|---|---|---|
| What it describes | how far single values spread | how much the average wobbles between samples | how far a poll figure could be from the truth | where the middle half of the values lies |
| Usual formula | s, with the n - 1 divisor | s / √n | 1.96 × SE for 95% | Q1 to Q3, by interpolation for grouped data |
| Shrinks with more data? | No | Yes, with √n | Yes, with √n | No |
| Always symmetric? | Yes, by construction | Yes, by construction | No, not for small counts | No |
| Where you meet it | lab reports, mean (SD) tables | journal tables, error bars | polls, surveys, A/B dashboards | census and textbook frequency tables |
The split in that row is the whole story in miniature. The standard deviation and the quartile range describe the people or things you measured, so measuring more of them only pins down the same spread more firmly. The standard error and the margin of error describe your estimate, and an estimate gets sharper with every observation you add.
Round one: standard deviation against standard error
Take 36 measurements with a mean of 103 and a standard deviation of 15, the scale of many standardized scores. The standard error of the mean is the standard deviation divided by the square root of the sample size: 15 / √36 = 2.50. Same data, two numbers six times apart, because √36 = 6.
The calculator prints the three plus-or-minus numbers that can be built from this one sample, and they describe three different things. Mean ± SD runs from 88.00 to 118.00 and covers roughly two thirds of the individual scores, if they are close to normal. Mean ± SE runs from 100.50 to 105.50 and says how much the average itself would move if you drew another 36 people. The 95% confidence interval, mean ± t × SE with t = 2.030 on 35 degrees of freedom, runs from 97.92 to 108.08.
Now add data and watch the two numbers part ways. At 144 measurements the standard error halves to 1.25; at 576 it is 0.63. The standard deviation stays near 15 throughout, because people did not become more alike just because more of them were measured. That is the test to apply to any unlabeled ±: if it would shrink with a bigger study, it is about the estimate; if it would not, it is about the individuals.
Error bars are where this round is most often lost. Two groups whose ±1 SE bars just touch differ by about two standard errors of each mean. With equal errors, the standard error of the difference is √2 times as large, so the gap is z = 2 / √2 = 1.41, a p-value of about 0.16. Touching SE bars are nowhere near significant, and non-overlapping ones are not automatically significant either. Journals that print "mean ± SEM" without saying so in the legend leave readers to guess which of the two worlds they are in.
The conversion runs both ways, which is useful when a paper reports only the standard error and you need the spread, say for a sample size plan or a meta-analysis. A reported 120 ± 2.5 (SEM) from 25 people means a standard deviation of 2.5 × √25 = 12.50. And if the paper gives only a 95% interval, say 4.2 to 7.8 from 25 people, the width divided by twice the t value (2.064 on 24 degrees of freedom) gives a standard error of 0.872 and a standard deviation of 4.361. The popular shortcut of dividing the width by 3.92 would give 0.918, about 5.3% too high, which is why the Cochrane Handbook recommends the t divisor for groups under 60.
Round two: standard error against the margin of error
These two are close relatives, and the relationship is one multiplication. The margin of error a poll publishes is the standard error of the percentage times the critical value for the chosen confidence level, 1.960 for 95%. For 1,000 respondents and a result near 50%, the standard error is √(0.5 × 0.5 / 1,000) = 1.58 points, and the margin is 1.960 × 1.58 = ±3.10 points. That is the "3.1 points" from the opening line, and "19 times out of 20" is just a friendlier way to say 95%.
So a poll reporting 52% for one side is really saying "somewhere between 48.90% and 55.10%, with 95% confidence". The lead over 50% is smaller than the margin, which is why careful reporters call such a race too close to call. The margin also depends on the answer itself. At 1,000 respondents it is ±3.10 near 50%, but only ±1.86 for an answer near 10%, because p(1 - p) peaks at a 50% split. Pollsters quote the 50% figure as the margin of the whole survey precisely because every other answer is at least that precise.
The margin shrinks with the square root of the sample size, so precision gets expensive quickly. Each row below opens the calculator with that sample size filled in:
| Respondents | Margin of error at 95%, 50% share |
|---|---|
| 100 | ±9.80 points |
| 250 | ±6.20 points |
| 300 | ±5.66 points |
| 400 | ±4.90 points |
| 500 | ±4.38 points |
| 1,000 | ±3.10 points |
| 2,000 | ±2.19 points |
| 4,000 | ±1.55 points |
Going from 1,000 to 4,000 respondents quadruples the cost and only halves the margin, from ±3.10 to ±1.55. That is why so many national polls stop near 1,000. The confidence level moves the margin too: the same 1,000 people give ±2.60 points at 90% and ±4.07 at 99%. A margin quoted without its confidence level is only half a number.
One thing the margin never covers is anything other than random sampling. People who refuse to answer, a sample drawn from the wrong list and a leading question can all move a result by more than 3 points, and none of that shows up in the plus-or-minus.
When the plus-or-minus stops being symmetric
The textbook margin, p ± z × SE, treats the uncertainty as the same in both directions. With large counts that is fine. With small ones it breaks. Two positive results out of 20 is a rate of 10%, and the textbook formula gives a 95% range of 0% to 23.15%, after cutting off a lower end that would have been below zero. The Wilson interval, which the confidence interval calculator uses as its headline for proportions, gives 2.79% to 30.10%, reaching 7.21 points below the sample rate and 20.10 above it. The exact Clopper-Pearson interval is wider still, 1.23% to 31.70%.
A symmetric ± around 10% would have to dip below zero on one side, which is impossible for a rate, and it understates the upper side, which is where the real risk sits. Whenever either count is small, a lopsided interval is the honest report, and a single plus-or-minus number is not.
Round three: standard deviation against the quartile range
The last pair both describe individuals rather than estimates, and neither shrinks with more data. The difference is in what they assume. The standard deviation weighs every value by its squared distance from the mean, so a few extreme values pull it hard, and it needs every value, or at least a midpoint for every class. The quartile range only asks where the lowest quarter ends and the highest quarter begins.
Heights of 80 students, grouped into six classes of 5 cm, make the comparison concrete. The grouped data quartile calculator interpolates inside the class that holds each quartile and returns Q1 = 161.11, a median of 166.25 and Q3 = 170.63 cm.
The middle half of the class is 9.51 cm wide, and half of that, the quartile deviation, is 4.76 cm. From the class midpoints, the same table has a mean of 165.88 cm and a standard deviation of 6.50 cm. For a normal distribution the interquartile range would be about 1.349 standard deviations, 8.77 cm; the table's 9.51 is a little wider, so the heights are spread more evenly across the middle than a bell curve would put them. Bowley's skewness, which compares the two halves of the middle band, is -0.080: close enough to zero to call the middle symmetric.
Where the quartile range wins outright is the table that ends in an open class. Commuting times are typically published as "under 20 minutes", then 10-minute bands, then "50 minutes or more". Take 60 commuters: 8 under 20 minutes, 15 from 20 to 30, 22 from 30 to 40, 10 from 40 to 50 and 5 at 50 or more. There is no midpoint for either open class, so the standard deviation cannot be computed without inventing one. The quartiles do not need it: Q1 = 24.67, median 33.18, Q3 = 40.00 minutes, because none of them falls in an open class. Income, age and wealth tables are built the same way, which is why official statistics report medians and quartiles far more often than means.
One trap is specific to grouped data. Textbooks disagree on whether the position of Q1 is N / 4 or (N + 1) / 4. For the 80 heights the two rules give 161.11 against 161.18 for Q1 and 170.63 against 170.86 for Q3. The gap shrinks as N grows, but on a worksheet it is the difference between a right and a wrong answer, so the calculator prints both.
The verdict: which range answers which question
None of the four is better than the others in general. Each is the right answer to one question and a misleading answer to the other three.
| The question | Print this | Switch when |
|---|---|---|
| How much do individuals differ? | Standard deviation | Quartile range if the data are skewed, grouped or have an open class |
| How precisely is the average known? | Confidence interval | Standard error only if the reader will do the multiplying |
| How far off could this percentage be? | Margin of error | Wilson or exact interval when either count is under about 5 |
| What is typical in a table of classes? | Median and quartiles | Mean and SD only if every class is closed |
Two of those verdicts deserve a word. The standard error loses to the confidence interval even though it carries the same information, because an interval can be read without arithmetic and a bare standard error invites exactly the error-bar mistake from round one. And the quartile range beats the standard deviation for skewed or grouped data not because it is more sophisticated, but because it asks less of the data: two cut points instead of a distance for every value.
Three habits for reading a range someone else printed
First, find the label. "Mean (SD)", "mean ± SEM" and "mean (95% CI)" are three different claims, and a legend or a methods section almost always says which one it is. If nothing says, be suspicious of a very tight range from a small study: it is probably a standard error.
Second, ask whether the range would shrink with a bigger study. If yes, it measures the estimate. If no, it measures the individuals. That single question sorts the four ranges into their two families without any formula.
Third, convert when you have to. The standard error times √n gives the standard deviation back; an interval's width divided by twice the t value gives the standard error; the poll margin divided by 1.96 gives the standard error of the percentage. With those three moves, any published plus-or-minus can be turned into the one your question actually needs.
Tools discussed in this article
- Confidence interval calculator: intervals for a mean from raw data or a summary, and for a proportion with Wilson, exact Clopper-Pearson, Agresti-Coull and Wald side by side; one-sided bounds, five confidence levels, the finite population correction, and the margin of error of a poll from its sample size alone.
- Standard error calculator: the standard error of a mean or a proportion from data, from an SD and n or from a count, the reverse conversions from a reported SEM or a published confidence interval, mean ± SD, ± SE and the 95% interval side by side, and the sample needed for a target precision.
- Grouped data quartile calculator: Q1, the median, Q3 and any percentile from a frequency table by interpolation, with class boundaries for gapped classes, open first and last classes, both position rules, the quartile deviation and Bowley skewness.
More statistics tools
Sample size (responses for a target margin) · Z-test (two rates or a rate against a benchmark) · Chi-square (tables of counts) · P-value (a p from any z, t, chi-square or F) · One-sample t-test (a mean against a target) · Two-sample t-test (two group means, Welch or paired) · Fisher's exact test (small 2x2 tables) · Z-score (one value against a mean and SD) · Wilcoxon test (rank tests for skewed data) · Pearson correlation (the r between two columns) · Linear regression (a line through paired data) · R squared (how much a model explains) · Skewness (how lopsided a list is) · Kurtosis (how heavy its tails are) · Interquartile range (quartiles from a raw list) · Coefficient of variation (spread as a share of the mean) · Percentile (where a value sits in a list) · Mode (the most frequent value) · Standard deviation (the spread of single values) · Median (the middle of a plain list)