---
url: "https://questionstar.com/blog/confidence-interval/"
title: "Confidence interval: how to calculate and interpret it"
description: "How to calculate and report a confidence interval for a proportion or mean, and what a 95% confidence interval means. With calculator and worked examples."
lastmod: "2026-11-12"
type: "blog"
---

# Confidence interval: how to calculate and interpret it

*How precise is my survey result, and how do I calculate and read a confidence interval?*

How to calculate and report a confidence interval for a proportion or mean, and what a 95% confidence interval means. With calculator and worked examples.

In short: a confidence interval is the range around a sample result in which the true value of the population (everyone you want to make a statement about) plausibly lies, for example 57.2% to 66.8% for a result of 62% among 400 randomly selected respondents. The 95% describes the method: of a very large number of intervals calculated this way, about 95 out of 100 contain the true value (Schumann 2019, pp. 191-193). The calculation requires a random sample: chance decides who is surveyed, not the respondents themselves (Jacob et al. 2019, p. 71). What that means for open links, and for surveys where everyone is invited and only some respond, is covered under [Assumptions](#assumptions).

**How to use the calculator.** Enter the number of valid responses to a question, and either the proportion in percent or the number of people who chose a particular answer. Choose the confidence level. 95% is the usual choice. The calculator shows the margin of error in percentage points, the interval and a ready-made sentence for your report. If fewer than ten people chose the answer, or fewer than ten did not choose it, it warns you, because the calculation becomes unreliable. The calculator works with proportions. For means, you'll find the formula, the calculation steps and an Excel formula under [Confidence interval for a mean](#mean).

How far can a result be from the true value? I'll go through five questions in turn:

1. [How do I calculate a confidence interval for a proportion?](#proportion)
2. [How do I calculate it for a mean?](#mean)
3. [What does the 95% mean, and what doesn't it mean?](#interpretation)
4. [What determines the width, and what can I do with small counts?](#width)
5. [How do I report the interval?](#reporting)

The example that runs through this article is made up, and it's the same one as in the article [P-value and statistical significance](/blog/statistical-significance/): an online store surveys 400 randomly selected customers, and 248 of them are somewhat or very satisfied with the service. That's 62%, the [top-2 box](/blog/rating-scale/#analysis) of this question. Whether two such values differ is the topic of the other article. This article is about how much confidence you can place in a single estimate.

## What is a confidence interval?

The 62% applies to the 400 respondents. But what the online store wants to know is how satisfied its entire customer base is. If chance had picked 400 other people, the report might say 59% or 65%. But how much can the result vary? You don't need to repeat the survey to find out: how much the result varies from sample to sample can be estimated from a single sample (Schumann 2019, p. 177). This quantity is called the **standard error**. It tells you how far the results of all conceivable samples of the same size typically lie from the true value (Malhotra and Birks 2007, p. 434). In the example, it's 2.43 percentage points.

A **confidence interval** (CI for short) is a range calculated from the sample in which the true value of the population plausibly lies. The intervals calculated here run from "result minus margin of error" to "result plus margin of error." At the usual 95%, the margin of error is 1.96 standard errors, because with large samples about 95 out of 100 sample results lie no farther than that from the true value. A range this wide around your result therefore includes the true value in about 95 out of 100 samples (Schumann 2019, p. 191). This 95% is called the confidence level. In the example: 1.96 × 2.43 = 4.8 percentage points, so 57.2% to 66.8%.

This tells you what you can conclude from the 62%: the data support the statement that more than half of the customer base is satisfied. Whether more than 60% are satisfied remains an open question.

| Term | Meaning | In the example (248 of 400) |
|------|------|------|
| Estimate | the result of the sample | 62% |
| Standard error | how much the result varies from sample to sample | 2.43 percentage points |
| Confidence level | share of the intervals calculated this way that contain the true value | 95% |
| Margin of error | half the width of the interval: a factor (about 2 at 95%) × standard error | ±4.8 percentage points |
| Confidence interval | estimate ± margin of error | 57.2% to 66.8% |

*Terms based on Schumann 2019, pp. 177 and 191, and Malhotra and Birks 2007, p. 434. Numbers: my own calculation.*

The words "percentage points" matter: what's meant is 62 plus or minus 4.8, not 4.8% of 62.

## How do I calculate a confidence interval for a proportion?

For a proportion p from n valid responses, the calculation is (Schumann 2019, p. 193; Malhotra and Birks 2007, pp. 440-441):

```math
\begin{aligned} \text{standard error } SE &= \sqrt{\frac{p \cdot (1 - p)}{n}} \\ \text{margin of error } E &= z \cdot SE \\ \text{interval} &= p - E \text{ to } p + E \\ z &= 1.645 \text{ at } 90\%,\ 1.96 \text{ at } 95\%,\ 2.576 \text{ at } 99\% \end{aligned}
```

> **Example.** For the 248 satisfied customers among 400 respondents, the formula gives a margin of error of ±4.8 percentage points, and the 95% confidence interval runs from 57.2% to 66.8%.
> 
> Step by step:
> 
> ```math
> \begin{aligned} p &= \frac{248}{400} = 0.62 \\ SE &= \sqrt{\frac{0.62 \cdot 0.38}{400}} = 0.0243 \\ E &= 1.96 \cdot 0.0243 = 0.0476\text{, that is } \pm 4.8 \text{ percentage points} \\ \text{95\% confidence interval} &= 57.2\% \text{ to } 66.8\% \end{aligned}
> ```

That's exactly what the calculator above does. When you plug in your numbers, keep this in mind:

- **n is the number of valid responses to this question**, not the number of invitations or of completed questionnaires. Anyone who skipped the question or chose an opt-out answer such as "can't judge" isn't counted.
- **For subgroups, use the number of valid responses in that subgroup.** For "62% of regular customers," use the number of regular customers surveyed.
- **In a multiple-choice question, each answer option is a proportion of its own.** n is the number of people who answered the question, not the number of boxes checked.
- **The NPS is not a simple proportion** but the difference between two proportions. The calculator isn't suitable for it.

### The margin of error depends on the proportion itself

The formula contains p × (1 − p). This product is largest at 50% and gets smaller toward the edges. With 400 responses, the margin of error for a proportion of 50% is ±4.9 percentage points, and for a proportion of 10% or 90% only ±2.9 (my own calculation, similar in Malhotra and Birks 2007, p. 432, for 385 respondents). The single margin of error "of the survey" that polling organizations usually state is the one for 50%, that is, the largest (AAPOR n.d., p. 1). For all other proportions, it's an upper limit. As a rule of thumb for 95%, you can use 1 / √n, so about ±5 percentage points with 400 responses ([slide 152](/principles-of-survey-research/chapter-4/slide-152.html)).

Relative to the proportion itself, though, the margin of error is large when the proportion is small: at 10% ±2.9 percentage points, the true proportion can be almost a third smaller or larger than measured.

When the formula stops working is covered under [small counts](#small-numbers).

## How do I calculate a confidence interval for a mean?

Many results aren't proportions but means, such as average satisfaction on a scale or average waiting time. The logic is the same, but the standard error is calculated differently. In place of √(p × (1 − p)), you use the standard deviation s of the responses (often SD in reports). It tells you how far the individual responses typically lie from the mean: if all responses are the same, it's 0, and the more the responses differ, the larger it gets. The calculation (Schumann 2019, pp. 191-192; Malhotra and Birks 2007, p. 435):

```math
\begin{aligned} \text{mean } \bar{x} &= \frac{\text{sum of all values}}{n} \\ \text{standard deviation } s &= \sqrt{\frac{\text{sum of the squared deviations from the mean}}{n - 1}} \\ \text{standard error } SE &= \frac{s}{\sqrt{n}} \\ \text{margin of error } E &= t \cdot SE \\ \text{interval} &= \bar{x} - E \text{ to } \bar{x} + E \\ t &= \text{a factor like } z\text{, close to } 1.96 \text{ at } 95\% \text{ with many responses, larger with few} \\ & \quad \text{(value of the t distribution with } n - 1 \text{ degrees of freedom)} \end{aligned}
```

For a mean, the factor t replaces 1.96. You don't have to calculate t yourself: values from the t distribution are tabulated for every number of responses, and a selection for 95% and the Excel formula are given below. You look up t under n − 1 degrees of freedom. Degrees of freedom tell you how many response values can vary independently: once the mean of n responses is fixed, only n − 1 values can be chosen freely. The last value is determined by the others. With few responses, t is larger than 1.96, because then the standard deviation is itself known only imprecisely, so you leave more room (more on this under "t or z").

You don't have to do this by hand. In Excel, AVERAGE(range) gives you the mean and STDEV.S(range) the sample standard deviation, which uses n − 1. You get the margin of error directly:

> ℹ️
> 
> In Excel: `=CONFIDENCE.T(0.05,STDEV.S(range),COUNT(range))`

The function returns the margin of error for 95%, calculated with t and n − 1 degrees of freedom, not the limits of the interval (Microsoft n.d.). So you subtract the margin of error from the mean and add it to the mean.

### Worked example: satisfaction on a scale from 1 to 5

> From 400 responses on a scale from 1 to 5, you get a mean of 3.61 with a margin of error of about ±0.11, and the 95% confidence interval runs from 3.50 to 3.72.
> 
> The online store asked:
> 
> > "Overall, how satisfied are you with our service?"
> > very dissatisfied (1) · somewhat dissatisfied (2) · neither satisfied nor dissatisfied (3) · somewhat satisfied (4) · very satisfied (5)
> 
> | Answer | Value | Count | Share |
> |------|------|------|------|
> | very dissatisfied | 1 | 20 | 5% |
> | somewhat dissatisfied | 2 | 44 | 11% |
> | neither satisfied nor dissatisfied | 3 | 88 | 22% |
> | somewhat satisfied | 4 | 168 | 42% |
> | very satisfied | 5 | 80 | 20% |
> | Total |  | 400 | 100% |
> 
> *Made-up example. "Somewhat satisfied" and "very satisfied" together give the 248 of 400 from the proportion example.*
> 
> For the interval, you need three numbers: the number of responses, the mean and the standard deviation. The standard deviation here is 1.08. That means the individual responses typically lie about one scale point from the mean. You'll find this number in the descriptive statistics of your report, or you can calculate it in Excel with STDEV.S. For the sake of completeness, the calculation is shown in the third line: 463.16 is the sum of the squared distances of all 400 responses from the mean, and 399 is n − 1.
> 
> ```math
> \begin{aligned} \text{sum of the values} &= 20 \cdot 1 + 44 \cdot 2 + 88 \cdot 3 + 168 \cdot 4 + 80 \cdot 5 = 1{,}444 \\ \bar{x} &= \frac{1{,}444}{400} = 3.61 \\ s &= \sqrt{\frac{463.16}{399}} = 1.08 \\ SE &= \frac{1.08}{\sqrt{400}} = \frac{1.08}{20} = 0.054 \\ t &= 1.966 \text{ (399 degrees of freedom, } 95\%\text{)} \\ E &= 1.966 \cdot 0.054 = 0.106\text{, that is about } \pm 0.11 \\ \text{95\% confidence interval} &= 3.50 \text{ to } 3.72 \end{aligned}
> ```
> 
> So the true mean of the customer base plausibly lies between 3.50 and 3.72.

Whether you may calculate a mean from five answer categories at all is disputed, and both sides are presented in the article [Levels of measurement](/blog/levels-of-measurement/#likert). If you report the mean of a single rating question, include the response distribution and the median: in this example, the table above and a median of 4. A confidence interval shows how precisely the mean has been estimated, but it doesn't make the mean more meaningful.

### t or z: when does it make a difference?

Strictly speaking, the formula for a mean should always use a t value rather than 1.96 (Schumann 2019, p. 192, n. 40). The reason: the standard deviation is itself only estimated from the sample, and with few responses this estimate is imprecise. The responses could in truth vary more than the sample shows. The larger factor t leaves more room for this additional uncertainty. The more responses you have, the more precisely the standard deviation is known and the closer t moves to 1.96. With 120 or more responses, the two are practically indistinguishable (Malhotra and Birks 2007, pp. 526-527).

| Responses n | t value at 95% | Interval wider than with 1.96 |
|------|------|------|
| 5 | 2.776 | by 42% |
| 10 | 2.262 | by 15% |
| 25 | 2.064 | by 5% |
| 30 | 2.045 | by 4% |
| 100 | 1.984 | by 1% |
| 400 | 1.966 | by 0.3% |

*My own calculation, t distribution with n − 1 degrees of freedom. Excel gives you the t value for your number of responses with =T.INV.2T(0.05,n−1).*

In small groups, the difference matters.

> **Example.** In the online store's sales team, 25 people responded, with a mean of 3.60 and a standard deviation of 1.08. The standard error is 1.08 / √25 = 0.216. With t = 2.064, you get ±0.45, so 3.15 to 4.05. With 1.96, it would be ±0.42, and the interval would look more precise than it is.

### When is the calculation for means reliable?

The calculation for means relies on the following conditions (Diez et al. n.d., sections 4.3 and 5.1):

- **Independent responses from a random sample.** More on this under [Assumptions](#assumptions).
- **Enough responses.** The means of many samples are approximately normally distributed, forming a bell-shaped curve: most are close to the true mean, and fewer lie farther away on either side. That holds even if the individual responses are distributed quite differently. The calculation rests on this shape. As a rule of thumb, this applies from about 30 responses (Bortz et al. 2010, p. 87, cited in Schumann 2019, p. 176). This threshold concerns the shape of the distribution, while the 120 from above concerns only the difference between t and z.
- **No strong skew, no outliers.** From about 30 responses, moderate skew is unproblematic, but strong skew or outliers call for caution (Diez et al. n.d., section 5.1). Monetary amounts and durations are typically skewed. In that case, check the distribution and report the median as well. With fewer than 30 responses on a scale with few points, the interval is a rough guide: for the 25 people in the sales team, "3.15 to 4.05" means little more than somewhere between "neither satisfied nor dissatisfied" and "somewhat satisfied."

## Interpreting a confidence interval: what does the 95% mean?

The confidence level is often misunderstood. It tells you how often the method used to calculate your interval captures the true value.

A thought experiment helps. Suppose exactly 60% of the online store's entire customer base were satisfied, and 20 teams each drew a random sample of 400 people, independently of one another. Each team would get a slightly different proportion and therefore a slightly different interval. The chart shows one such simulation.

19 of the 20 intervals contain the true value of 60%, and one misses it: sample 8 happened to include an unusually small number of satisfied customers, 217 of 400, and its interval ends at 59.1%. The team with sample 8 doesn't notice any of this.

The simulation was chosen so that exactly one interval misses. With 20 samples, it can also be none or two. In the long run, with a very large number of samples, about 95 out of 100 intervals contain the true value (Schumann 2019, p. 191; Diez et al. n.d., section 4.3). That's what "95% confidence level" means.

Here's how you can remember it and say it: the method captures the true value in 95 out of 100 cases. So we can be 95% confident that the true value lies between 57.2% and 66.8%.

In an academic paper, you word it more strictly: the 95% describes the success rate of the method, not the probability for this one interval (Schumann 2019, p. 191). The reason is the distinction between what can be said before and after the survey (Schumann 2019, p. 191 with n. 37):

- **Before the survey**, the statement is a probability: with a probability of 95%, the interval you'll get will enclose the true value.
- **After the survey**, your interval is fixed, and the true value is also a fixed number, just an unknown one. It lies in your interval or it doesn't. Strictly speaking, a probability of 95% can no longer be stated for that.

It's similar with a ticket drawn from a raffle drum in which 95 of 100 tickets are winners: the ticket you've drawn is already a winning or a losing ticket, even if you haven't opened it yet, and your 95% confidence rests on how the drum is filled. The 95% describes the drum, not the single ticket.

### Five readings that aren't correct

| Statement | Is that right? | What is correct |
|------|------|------|
| "95% of the respondents lie within the interval." | No | The interval concerns the value of the population, not individual responses (Diez et al. n.d., section 4.3). In the example, the interval for the mean runs from 3.50 to 3.72. Not a single one of the 400 responses lies within it, because people answered with 1, 2, 3, 4 or 5. |
| "If the survey is repeated, there's a 95% probability that the new result will lie in my interval." | No | The interval says nothing about the results of other samples, that is, about their estimates (Diez et al. n.d., section 4.3). The new result varies as well and has its own interval. |
| "The interval covers all the uncertainties of the survey." | No | It captures only random sampling error. Questions that are easy to misunderstand, nonresponse or an incomplete address list are non-sampling errors and aren't included in the interval (Malhotra and Birks 2007, p. 85). |
| "With 99% instead of 95%, the result gets better." | No | Only the interval gets wider. More certainty costs precision, and the data stay the same (Schumann 2019, p. 192). |
| "A narrow interval shows that the result is right." | No | A narrow interval means a small calculated random error, not an unbiased selection. The sheer size of a sample is no indicator that it represents the population correctly (Jacob et al. 2019, p. 71). |

*My own compilation based on the sources named.*

In a report, you're on the safe side if you state the interval and its level: "The 95% confidence interval runs from 57.2% to 66.8%." For an audience without a background in statistics, "The true proportion plausibly lies between 57% and 67%" works as well.

### 90, 95 or 99%?

A parcel service announces delivery "between 8 a.m. and 8 p.m." With that, it's almost always right. But that doesn't help you much: the parcel can come at 9 a.m., at 3 p.m. or at 7 p.m., and you'd have to stay home all day. If it promises "between 2 and 4 p.m.," you can plan. But it can keep this promise only if it knows more: where the driver is right now and when the driver was on your street over the past few days. The more such observations it has, the more precisely it can say when the driver usually reaches you. Without them, it misses more often with the narrow time window.

A confidence interval works in a similar way. A wide interval is like the time window from 8 a.m. to 8 p.m.: you can be very sure that the true value lies within it, but your statement isn't very specific, because the value can lie almost anywhere. A narrow interval is more specific, like the time window from 2 to 4 p.m. But the narrower you draw it, the more easily the true value falls outside it. If you want a narrow interval and still want to be sure, you have to know more, like the parcel service. Its observations are your responses in a survey: the more responses you have, the more precisely you know where the true value lies.

Often, though, the number of responses is fixed, because the survey is finished. Then what's left is a trade-off. You can be more sure, and then the interval gets wider. Or you can be more specific, and then your certainty drops. More of one means less of the other. You make this choice by setting the confidence level: it states how sure you want to be. 95% is usual, and depending on the research question 99% or 99.9% is used as well (Schumann 2019, p. 192). Survey research also works with [90%](/blog/calculating-sample-size/#fehlerspanne-ansatz).

The table shows both for our example, in which 62% of respondents are satisfied: in the first three rows the trade-off with 400 responses, and in the last one what more responses achieve.

| Confidence level | Responses | Margin of error | Confidence interval |
|------|------|------|------|
| 90% | 400 | ±4.0 percentage points | 58.0% to 66.0% |
| 95% | 400 | ±4.8 percentage points | 57.2% to 66.8% |
| 99% | 400 | ±6.3 percentage points | 55.7% to 68.3% |
| 95% | 1,600 | ±2.4 percentage points | 59.6% to 64.4% |

*My own calculation for a proportion of 62%.*

In short: a wide interval is safe, but not very specific. A narrow one is specific, but riskier. You get both together only with more responses. And the confidence level is the name for the choice you make between the two.

Set the level before the analysis, and always state it. An interval without a level can't be interpreted.

## What determines the width?

Three quantities determine how wide a confidence interval gets (Schumann 2019, p. 192):

- **The number of responses.** More responses make the interval narrower.
- **The confidence level.** More certainty makes it wider.
- **The variability.** The more the responses differ, the wider the interval. For means, the variability is in the standard deviation, and for proportions in the product p × (1 − p).

In practice, the only one of these you can influence is the number of responses. The variability is what it is, and the confidence level is set by convention ([slide 153](/principles-of-survey-research/chapter-4/slide-153.html)). And the number of responses works only through its square root: for an interval half as wide, you need about four times as many responses (Schumann 2019, p. 192, n. 43).

| Responses n | Proportion 62%: margin of error | Mean 3.61 (s = 1.08): margin of error |
|------|------|------|
| 50 | ±13.5 percentage points | ±0.31 |
| 100 | ±9.5 percentage points | ±0.21 |
| 400 | ±4.8 percentage points | ±0.11 |
| 1,600 | ±2.4 percentage points | ±0.05 |

*My own calculation, 95%, simple random sample, large population.*

In practice, this means: the total sample of 400 doesn't help the sales team with its 25 responses. For a proportion of 60%, the margin of error there is ±19.2 percentage points. The article [P-value and statistical significance](/blog/statistical-significance/#confidence-interval) has a chart on this.

If you want to go the other way, setting a margin of error and working out the number of responses you need, the article [Calculating sample size](/blog/calculating-sample-size/) and the [sample size calculator](/blog/sample-size-calculator/) will help.

## What to do with small counts and small populations?

The formula for proportions is an approximation based on the normal distribution. It becomes unreliable when only a few people chose an answer or did not choose it. The calculator above warns according to the widely used [rule of ten](/blog/statistical-significance/#when-the-formula-stops-working): at least 10 people should have given the answer and at least 10 should not have (Diez et al. n.d., section 6.1).

> **Example.** With 3 of 30 respondents, the formula gives ±10.7 percentage points, with calculated limits of −0.7% and 20.7%, and only about 81 out of 100 such intervals contain a true proportion of 10%.
> 
> In detail: 3 of 30 respondents reported a problem. That's 10%. The formula gives ±10.7 percentage points, so the calculated interval runs from −0.7% to 20.7%. A negative proportion is impossible, so the calculator shows 0.0% as the lower limit. And the method no longer keeps its promise: if the true proportion is 10%, then with 30 respondents only about 81 out of 100 intervals calculated this way contain that value, not 95 (my own calculation).

Brown, Cai and DasGupta showed that the coverage of the usual formula, that is, the share of intervals that contain the true value, fluctuates erratically and that common textbook rules don't reliably protect against this (Brown et al. 2001, p. 101). So the rules of thumb are a minimum check, not a guarantee. Statistics software offers more accurate methods for such cases. For small samples, Brown and his coauthors recommend the Wilson interval, which never extends below 0% or above 100% (Brown et al. 2001, pp. 107 and 115).

When you report very small counts, frequencies are often more honest than percentages. "3 of 30 respondents" says everything, while "10%" suggests a precision that isn't there. The same goes for thinly populated cells in a [crosstab](/blog/cross-tabulation/).

### Small population: the finite population correction

All the formulas so far assume that the population is much larger than the sample. If you survey a large part of a small group, you know more than the formula assumes, and the interval comes out too wide. A correction should be applied when the sample includes 10% or more of the population. The correction factor is √((N − n) / (N − 1)), where N is the size of the population (Malhotra and Birks 2007, pp. 433 and 435). You multiply the margin of error by it, for proportions and for means alike.

> **Example.** In an association with 250 members, 150 of whom were surveyed, the correction reduces the margin of error from ±7.84 to ±5.0 percentage points: 55.0% to 65.0% instead of 52.2% to 67.8%.
> 
> In detail: the association has 250 members, 150 randomly selected members were surveyed, and 60% agree with a proposal.
> 
> ```math
> \begin{aligned} \text{without correction: } E &= 1.96 \cdot \sqrt{\frac{0.60 \cdot 0.40}{150}} = 0.0784\text{, that is } \pm 7.84 \text{ percentage points} \\ \text{correction factor} &= \sqrt{\frac{250 - 150}{250 - 1}} = 0.634 \\ \text{with correction: } E &= 7.84 \cdot 0.634 = 4.97\text{, that is } \pm 5.0 \text{ percentage points} \\ \text{interval} &= 55.0\% \text{ to } 65.0\% \text{ instead of } 52.2\% \text{ to } 67.8\% \end{aligned}
> ```

The calculator above doesn't apply the correction. It only alerts you to the need for it. The derivation is in the article [Calculating sample size](/blog/calculating-sample-size/#endliche-grundgesamtheit).

The correction applies only if the respondents were drawn at random from the small group. If you invite all 250 members and 150 respond, that's no longer a random sample. More on this under [Assumptions](#assumptions).

## Comparing two values: the interval of the difference

Often what matters isn't a single value but a difference: this year against last year, department A against department B. Then you report the confidence interval of the difference. For two proportions from independent samples, the uncertainties of both values add up under the square root (Diez et al. n.d., section 6.2). The interval of the difference is therefore wider than either of the two single intervals. The calculation for the example, what the p-value says about it and why two overlapping single intervals aren't a test are covered in the article [P-value and statistical significance](/blog/statistical-significance/#comparison). For comparing two means, [the t-test](/blog/the-t-test/) is the right tool.

## Does this apply to every survey? The assumptions

A confidence interval rests on the assumption that chance decided who is in the sample: such intervals can be calculated only for randomly selected units (Jacob et al. 2019, p. 71). The American professional association AAPOR also states that the margin of error applies only to random samples and not to online surveys with self-selection (AAPOR n.d., p. 1).

You should distinguish four cases:

- **A random sample from a list**, for example drawn from the customer database and invited personally: the assumption is met in principle, as long as the list covers the population and nonresponse doesn't distort the picture.
- **An open link**, for example on social networks or on the website: the participants selected themselves. Describe the patterns you see in the surveyed group and don't report margins of error. More on this under [Representative sample](/blog/representative-sample/).
- **Everyone invited, some respond**, for example in an employee survey: who responds isn't decided by chance. The interval then describes only the random error under the assumption that those who responded don't differ systematically from the others ([response rate](/blog/survey-response-rate/)).
- [**Stratified samples](/blog/sampling-methods/#stratified-random-sampling), [cluster samples](/blog/sampling-methods/#cluster-sampling) or weighted data:** the calculation has to match the sampling method. The simple formulas on this page apply to the simple random sample.

And even with a properly drawn random sample, the interval measures only random sampling error, not the quality of the question: a leading question yields a precise interval around the wrong value.

## How do I report a confidence interval?

A fully reported result includes four pieces of information: the value, the interval or the margin of error, the confidence level and the number of responses. In addition, state the type of sample once in the report.

| Case | Wording |
|------|------|
| Proportion, random sample | "62% of respondents are somewhat or very satisfied (95% confidence interval 57.2% to 66.8%, n = 400)." |
| Proportion, short form | "62% satisfied (±4.8 percentage points at a 95% confidence level, n = 400)" |
| Mean | "Mean satisfaction is 3.61 on a scale from 1 to 5 (95% confidence interval 3.50 to 3.72, SD = 1.08, n = 400)." |
| Small group | "In the sales team, 15 of 25 respondents are somewhat or very satisfied. With so few responses, the proportion can only be estimated roughly." |
| Very small count | "3 of 30 respondents reported a problem." |
| Small population | "60% agree (95% confidence interval 55.0% to 65.0%, 150 of 250 members surveyed, with the finite population correction)." |
| Difference | "The proportion is 62%, up 4 percentage points from 58% last year (95% confidence interval for the difference −2.8 to +10.8 percentage points, 400 respondents in each sample)." |
| Self-selection, open link | "The results should be read as indicative of the surveyed group only. Margins of error are not reported, as they require random selection." |

*My own wording. The last row comes from the article [Representative sample](/blog/representative-sample/).*

A few more things help when reporting:

- **Round sensibly.** Whole percentages or one decimal place are usually enough. More digits don't make the result more reliable.
- **Label error bars.** Say in the legend what the bar shows, for example "Error bars: 95% confidence interval."
- **Explaining it once is enough.** In a report with many percentages, a methods note at the beginning will do: "With 400 respondents, the margin of error for proportions is at most ±4.9 percentage points (95% confidence level). For subgroups, it's larger."

## Confidence intervals for your QUESTIONSTAR data

For every question, the reports in QUESTIONSTAR show the count and percentage for each answer option ([Features](/features/)). Descriptive statistics can be switched on in practically every report: number of responses, mean, standard deviation, variance, minimum, maximum and quartiles. The reports do not show confidence intervals for either proportions or means. With the numbers from the report, you calculate the interval yourself.

**Proportion.** Take the number of valid responses and the proportion from the report and enter both in the calculator above. For multiple-choice questions, the percentage base is a setting in the report: the percentages refer either to the people who answered the question or to all boxes checked. For the calculator, you need the first base, so select that option in the report.

**Mean.** In four steps:

1. Switch on the descriptive statistics in the report and read off the number of responses, the mean and the standard deviation.
2. Divide the standard deviation by the square root of the number of responses. That's the standard error.
3. Multiply it by the t value for your number of responses ([table and Excel formula](#mean)). That's the margin of error.
4. The mean minus and plus the margin of error gives the interval, in the example 3.61 ± 0.11, so 3.50 to 3.72.

With the numbers of responses typical of surveys, it makes no difference to this calculation whether the standard deviation was calculated with n or with n − 1 in the denominator. If you prefer to work with the raw data, export them (SPSS, CSV or XLSX) and use the CONFIDENCE.T formula shown there in Excel.

If you work with an AI assistant connected to QUESTIONSTAR via the [MCP interface](/help/integrations/mcp-ai/), you can ask it: "For question 5, calculate the 95% confidence interval for each answer proportion and for the mean, using the t distribution for the mean." The calculation is then done by the assistant, not by QUESTIONSTAR. Have it name the formula and the number of responses, and check one result with the calculator above. One exception is the paired t-test: QUESTIONSTAR calculates it server-side and returns the mean difference with its 95% confidence interval. Questions you've marked as personal data are not passed to the assistant when the corresponding setting is switched on.

You create the most important condition before the survey: draw a random sample from your list and send each selected person a personal email invitation instead of sharing an open link. Only then does the interval mean what it promises.

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## Checklist: calculating and reporting a confidence interval correctly

- Type of sample clarified: random sample (from which list) or self-selection. With self-selection, no margin of error reported
- Calculated with the number of valid responses per question, and for subgroups with the number of cases in the subgroup
- Proportion: at least 10 people chose the answer and 10 did not, otherwise counts reported ("3 of 30") or a more accurate method from statistics software used
- Mean: calculated with t instead of 1.96, distribution checked and shown, and labeled as a rough guide with fewer than 30 responses, strong skew or outliers
- Sample covers 10% of the population or more: finite population correction applied
- Confidence level set before the analysis
- Value, interval or margin of error, level and n reported together
- Wording chosen to suit the audience: the strict version in academic papers (95% as the success rate of the method), values rounded sensibly
- For comparisons, the interval of the difference reported, not two single intervals placed side by side

## Frequently asked questions about confidence intervals

### What is a confidence interval, in simple terms?

A confidence interval is a range around a result, calculated from the sample, in which the true value of the population plausibly lies. With 62% satisfied among 400 randomly selected respondents, the 95% confidence interval runs from 57.2% to 66.8%. It shows how precisely the sample estimates the value (Schumann 2019, pp. 191-193).

### How do I calculate a confidence interval for a mean?

Standard error = standard deviation / square root of n, margin of error = t × standard error, interval = mean ± margin of error. t is the value of the t distribution with n − 1 degrees of freedom, and at 95% with many responses it's close to 1.96 (Schumann 2019, pp. 191-192; Malhotra and Birks 2007, pp. 526-527). Example: a mean of 3.61, a standard deviation of 1.08 and 400 responses give ±0.11, so 3.50 to 3.72.

### What does a 95% confidence interval mean?

The method captures the true value in 95 out of 100 cases: of a very large number of intervals calculated this way, about 95 out of 100 contain the true value. So you can be 95% confident that the true value lies in your interval. Strictly speaking, the 95% describes the success rate of the method, not the probability for this one interval (Schumann 2019, p. 191).

### What do I do if fewer than ten people chose an answer?

Then the usual formula for proportions is unreliable (Diez et al. n.d., section 6.1), and the calculated interval can even extend below 0%. Report counts instead of percentages, for example "3 of 30 respondents," or use a more accurate method from statistics software, such as the Wilson interval (Brown et al. 2001, p. 115).

### How do I get a narrower confidence interval?

In practice, only with more responses. For half the width, you need about four times as many (Schumann 2019, p. 192, n. 43). How many responses you need for a desired margin of error is shown in the article [Calculating sample size](/blog/calculating-sample-size/).

### Can I report a confidence interval for an online survey with an open link?

Better not. A confidence interval requires random selection (Jacob et al. 2019, p. 71). If the participants selected themselves, the margin of error doesn't apply (AAPOR n.d., p. 1). In that case, describe the results as patterns in the surveyed group and explain how the participants were recruited.

## Sources

AAPOR, American Association for Public Opinion Research (n.d.). Margin of Sampling Error/Credibility Interval. Statement, [aapor.org (PDF)](https://aapor.org/wp-content/uploads/2023/01/Margin-of-Sampling-Error-508.pdf), accessed November 11, 2026. p. 1 (margin of error only for random samples, largest at 50%).

Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval Estimation for a Binomial Proportion. Statistical Science, 16(2), 101-133. DOI 10.1214/ss/1009213286. Freely available in the [repository of the University of Pennsylvania](https://repository.upenn.edu/handle/20.500.14332/47517), accessed November 11, 2026. p. 101 (weaknesses of the usual approximate interval), p. 107 (Wilson interval), p. 115 (recommendation for small samples).

Diez, D. M., Barr, C. D. and Çetinkaya-Rundel, M. (n.d.). OpenIntro Statistics. Online edition at LibreTexts (stats.libretexts.org), no edition stated, accessed November 11, 2026. Section 4.3 "Confidence Intervals" (interpretation, conditions for means), section 5.1 "One-Sample Means with the t Distribution" (t interval), section 6.1 "Inference for a Single Proportion" (at least 10 successes and 10 failures), section 6.2 "Difference of Two Proportions" (standard error of the difference).

Jacob, R., Heinz, A. and Décieux, J. P. (2019). Umfrage. Einführung in die Methoden der Umfrageforschung [Surveys: an introduction to the methods of survey research] (4th ed.). Berlin/Boston: De Gruyter Oldenbourg. In German. p. 71 (random selection, confidence intervals).

Malhotra, N. K. and Birks, D. F. (2007). Marketing Research. An Applied Approach (3rd European edition). Harlow: Pearson Education. p. 85 (total error), p. 432 (margin of error by proportion), pp. 433-435 (sampling distribution, standard error, finite population correction), pp. 437-441 (confidence intervals for means and proportions), pp. 526-527 (t distribution).

Marx, P. (2026). Principles of Survey Research. Online course, chapter 4, slides 152 and 153. QUESTIONSTAR. [Slide 152](/principles-of-survey-research/chapter-4/slide-152.html), [slide 153](/principles-of-survey-research/chapter-4/slide-153.html).

Microsoft (n.d.). CONFIDENCE.T function. Excel help page, [support.microsoft.com](https://support.microsoft.com/en-us/office/confidence-t-function-e8eca395-6c3a-4ba9-9003-79ccc61d3c53), accessed November 12, 2026 (syntax and worked example).

Schumann, S. (2019). Repräsentative Umfrage. Praxisorientierte Einführung in empirische Methoden und statistische Analyseverfahren [Representative surveys: a practical introduction to empirical methods and statistical analysis] (7th ed.). Berlin/Boston: De Gruyter Oldenbourg. In German. pp. 176-177 (central limit theorem, standard error), pp. 191-193 (confidence intervals for means and proportions).

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I hope your results give you more than a number: a clear sense of how far you can trust it.

**Related articles:**

- [P-value and statistical significance, explained simply](/blog/statistical-significance/)
- [Calculating sample size: technical details explained simply](/blog/calculating-sample-size/)
- [Sampling methods: random, quota and cluster sampling](/blog/sampling-methods/)

*Dr. Paul Marx*

Author: Dr. Paul Marx
Dr. Paul Marx is the founder and Managing Director of QUESTIONSTAR and was a professor of marketing at the University of Siegen.

## Related articles

- [statistical-significance](/blog/statistical-significance)
- [calculating-sample-size](/blog/calculating-sample-size)
- [sampling-methods](/blog/sampling-methods)

*Author: Dr. Paul Marx — see [About](/about).*