Percentages in a dissertation – share of a sample and the confidence interval

The response rate and sample shares

The response rate tells you what proportion of those targeted replied: respondents / sample × 100. If a survey was sent to 240 people and 87 replied, the response rate is 36.3%. This figure should always be reported, because it affects how far the results can be generalised.

When you report shares, also give n, the number of observations. "62% of respondents (n = 54) were satisfied" is informative; a bare "62% were satisfied" doesn't say whether that's fifty respondents or five. In small samples a single respondent can shift the percentage by several units, so it's best not to show more than one decimal.

The confidence interval – how precise is a percentage

A percentage worked out from a sample is an estimate of the population's share. Its precision is described by a confidence interval. For a simple random sample the standard error of a share is

where p is the share as a decimal and n is the sample size. The most commonly used 95% confidence interval is found by adding and subtracting about 1.96 × SE. The calculator does this for you.

In your dissertation, use the formula and confidence level that your own methods literature specifies, and cite the source. If the sampling isn't a simple random sample – for example stratified, cluster or a convenience sample – the formula above underestimates the error.

Per cent or percentage point in the report

This is the most common wording error in the results section of a dissertation. When the two numbers being compared are themselves percentages, their difference is in percentage points.

• Correct: "The share of satisfied respondents rose from 34.5% to 41.2%, i.e. 6.7 percentage points."
• Correct: "The share of satisfied respondents rose by 19.4% in relative terms."
• Wrong: "The share of satisfied respondents rose by 6.7 per cent."

Both correct wordings describe the same change, but they answer different questions. Choose the one that serves the reader, and stay consistent throughout the work.

Rounding, totals and tables

1. Round only at the end. Rounding intermediate steps compounds the error.
2. Decimals consistently. One decimal is usually enough; use two only if the sample is large.
3. Shares don't always add up to a hundred. Rounding causes 99.9% or 100.1%. Add a note to the table; don't change the figures by hand.
4. Report missing answers. Say whether you calculated the percentages from the whole sample or only from those who answered the question – these give a different figure.
5. Use the percent sign correctly. In UK English it's written closed up to the number: 41.2%.

When you move the data into a spreadsheet, you'll find ready formulas on the Excel percentage formulas page. The basic formulas and exercises are on the percentage formulas page.

Cross-tabulation and comparing shares

When you compare two groups, work out the percentages within the group, not from the whole dataset. If 120 of the respondents are women and 80 are men, "45% of the women were satisfied" is comparable, but "27% of all respondents were satisfied women" tells you nothing about the difference between the groups.

Always mark in the table which way the percentages were calculated: as row or column percentages. The same number can look completely different depending on the direction, and the reader can't tell from the figures alone. Practical guidance for cross-tabulation in Excel is in the pivot section of the Excel percentage formulas.

Professional calculation tasks

Some students arrive on this page looking for calculations in their own field – for example concentrations in medication calculations, or shares of laboratory values. The percentage maths is the same, but the units, dilutions and safety requirements are not.

This site does not provide drug dosages, interpretations of laboratory results or treatment recommendations. For medication calculations, use your institution's teaching material and your workplace's guidance, and verify every calculation as instructed. For questions about your health, consult a healthcare professional.

The basics of research percentages

A percentage from a sample is an estimate of the population. That's why it should be reported alongside the number of observations and often a confidence interval.

Formulas and examples

Response rate

Example

87 replied out of 240: 36.3%.

Note

Always report n, the number of observations, as well.

Standard error of a share

Example

p = 0.363 and n = 240: standard error about 3.1 percentage points.

Note

p is the share as a decimal, not a percentage.

95% confidence interval

Example

In the example above, about 30.2 – 42.4%.

Note

The margin of error is largest when the share is near 50%.

Percentage point in the report

Example

34.5% → 41.2% is 6.7 percentage points, 19.4% in relative terms.

Note

Always say in the text which figure you're using.

Concepts and classification

Limitations of the calculator

• The formula assumes a simple random sample; other sampling designs have a larger error.
• In small samples a single respondent changes the percentage by several units.
• The calculator does not run statistical significance tests.
• In a dissertation, use the method your source literature specifies.

How up to date the information is

When it was checked

Content and the formulas used were checked on 12 August 2026.

Disclaimer

The calculator gives a mathematical result from the numbers you enter. It is not an official decision, an offer or professional advice. See the terms of use.

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