Standard deviation of Likert scale data
Survey software will happily print a mean and standard deviation for every 1–5 question. The numbers are easy to compute. Whether they mean anything depends on what the 1 to 5 stands for, and on whether you are looking at a single item or a score built from several.
━ Item A (mean 3.00, SD 0.65) ━ Item B (mean 3.00, SD 1.78)
The problem: the numbers are labels in order
A Likert item asks people to pick one of a fixed set of ordered answers, typically "strongly disagree", "disagree", "neutral", "agree" and "strongly agree", coded 1 to 5. The format is named after the psychologist Rensis Likert, who described it in 1932. The codes tell you the order of the answers. They do not tell you that the step from "disagree" to "neutral" is the same size as the step from "agree" to "strongly agree". In the terms of S. S. Stevens's 1946 classification of measurement scales, the data is ordinal, not interval.
The standard deviation needs interval data. It is built from distances to the mean (see the standard deviation formula), and a distance of 2 only means the same thing everywhere on the scale if the steps are equal. For a single ordinal item, an SD of 1.2 is a number computed from labels. It is not wrong arithmetic, but it rests on an assumption the question design never promised.
Two items with the same mean
The practical danger is not philosophical. It is that the mean and SD can hide the one thing a survey reader wants to know: do people agree with each other? Take two hypothetical items, each answered by 20 people:
| Response | Item A | Item B |
|---|---|---|
| 1 Strongly disagree | 0 | 7 |
| 2 Disagree | 4 | 2 |
| 3 Neutral | 12 | 2 |
| 4 Agree | 4 | 2 |
| 5 Strongly agree | 0 | 7 |
Both items have a mean of exactly 3.00. For item A the sum is 4 × 2 + 12 × 3 + 4 × 4 = 60; for item B it is 7 × 1 + 2 × 2 + 2 × 3 + 2 × 4 + 7 × 5 = 60. Both give 60 / 20 = 3. Now the spread, using the sample formula:
Item A: Σ(x − 3)² = 4 × 1 + 12 × 0 + 4 × 1 = 8 s = √(8 / 19) = 0.65 Item B: Σ(x − 3)² = 7 × 4 + 2 × 1 + 2 × 0 + 2 × 1 + 7 × 4 = 60 s = √(60 / 19) = 1.78
Here the SD does its job. A mean of 3.00 on its own suggests a room full of shrugs, and that is true of item A. Item B is a room split into two camps, with 14 of 20 people at one extreme or the other. Reporting "mean 3.0" without a spread measure would hide the most important finding of the survey, and an SD of 1.78 against 0.65 flags it at once.
So the SD is not useless on Likert data. It is a reasonable warning light for polarisation. What it is poor at is telling you what the polarisation looks like, and the ordinal summaries below do that better.
Try it: standard deviation calculator
The calculator below holds item B's 20 answers as codes 1 to 5; replace them with the coded responses to one of your own survey items.
Separate numbers with commas, spaces or new lines, or paste a spreadsheet column.
Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.
Your results will appear here: the standard deviation first, then the rest of the summary and a chart.
Standard deviation (sample)
1.77705
Your values typically sit about 1.78 above or below their mean of 3, in the same units as your data. 6 of 20 values (30%) fall between 1.223 and 4.777, within one standard deviation of the mean; for normally distributed data about 68% would.
Population SD (σ): 1.73205, if these values are the whole group.
- Count (n)
- 20
- Mean (x̄)
- 3
- Variance (s²)
- 3.15789
- Standard error
- 0.39736
- Minimum
- 1
- Q1 (25%)
- 1
- Median
- 3
- Q3 (75%)
- 5
- Maximum
- 5
- Range
- 4
More statistics (5)
- Relative SD (%RSD)
- 59.2349%
- Coefficient of variation
- 0.592349
- Sum (Σx)
- 60
- Sum of squares, Σ(x − x̄)²
- 60
- IQR (Q3 − Q1)
- 4
Data distribution
Shaded bands mark ±1, ±2 and ±3 SD from the mean. 6 of 20 values (30%) fall within ±1 SD.
Chart as text
Mean 3, sample standard deviation s = 1.77705, from 20 values between 1 and 5.
- Within ±1 SD (1.223 to 4.777): 6 of 20 values (30%). About 68% for normal data.
- Within ±2 SD: 20 (100%). About 95% for normal data.
- Within ±3 SD: 20 (100%). About 99.7% for normal data.
Show the working, step by step
Open the full standard deviation calculator for population mode, other examples and the step-by-step working.
What to report instead, or alongside
| Summary | Item A | Item B |
|---|---|---|
| Mean | 3.00 | 3.00 |
| Standard deviation (sample) | 0.65 | 1.78 |
| Median | 3 | 3 |
| Interquartile range (Q1 to Q3) | 3 to 3 | 1 to 5 |
| % agree (4 or 5) | 20% | 45% |
| % disagree (1 or 2) | 20% | 45% |
The median matches the mean for both items, so it does not separate them either. The interquartile range does, and it only uses the order of the answers: for item A the middle half of respondents all said "neutral", while for item B the middle half runs from one end of the scale to the other. (The quartiles here are the same under the common inclusive and exclusive methods.)
The clearest summary for a non-statistical reader is usually the pair of percentages. "45% agree and 45% disagree" says what "SD 1.78" only hints at. Better still, show the five counts as a stacked bar, which takes one line of a report and hides nothing.
When the mean and SD are reasonable
Many surveys combine several items that measure the same thing into a scale score: ten questions about job satisfaction, say, summed or averaged into one number per person. A sum of ten 1–5 items runs from 10 to 50 in steps of one, has many more possible values, and tends to behave much more like interval data. Treating it with the mean and SD is standard practice in psychology and education, and the psychology guide covers how those SDs are usually reported.
A rough rule for choosing:
| Data | Safe summary | Mean and SD? |
|---|---|---|
| One Likert item | Counts, median, IQR, % agree | As a secondary summary, alongside the counts |
| Scale score from several items | Mean and SD | Yes, usual practice |
| 0–10 rating (e.g. likelihood to recommend) | Median and IQR, or the distribution | Often reported; check for clumping at the ends |
| Comparing two groups on one item | Distribution per group; rank-based test | Only with the distributions shown |
The ceiling on the SD
A 1–5 scale also limits how large the SD can get, which makes it harder to compare with SDs from other scales. The most spread-out answer pattern possible is half the respondents at 1 and half at 5. With 20 people that gives a population SD of exactly 2 and a sample SD of √(80 / 19) = 2.05. Item B, at 1.78, is close to that ceiling. On a 1–7 scale the ceiling is higher, so an SD of 1.5 means something different on a seven-point item than on a five-point one. The coefficient of variation does not rescue this, because a Likert scale has no true zero; compare SDs only between items with the same number of answer options.
If you do report means for single items, which is common in course evaluations and product ratings, the average rating calculator works directly from the counts in each category. Put the SD next to the mean, and the counts next to both.
Related calculators
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IQR calculator
The middle 50% of responses, the ordinal-friendly spread.
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Median calculator
The middle answer, with no assumptions about spacing.
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Standard deviation in psychology
How SDs are used and reported with scale scores.
Common questions
What is a high standard deviation on a 1–5 Likert item?
The largest possible sample SD on a 1–5 item is a little over 2 (2.05 with 20 responses, when half choose 1 and half choose 5). An SD around 0.6 means most people gave the same or a neighbouring answer; one around 1.5 or more usually means opinion is split. Look at the response counts before drawing that conclusion.
Can I use a t-test on Likert data?
For a single item the Mann–Whitney U test is the more defensible choice, because it uses only the order of the answers. For a scale score built by summing several items, a t-test is widely used and generally behaves well. Whichever you choose, report it, and show the response distribution.
Should I report the mean or the median for a Likert item?
The median is the safe default for a single item, with the interquartile range or the percentage who agree. The mean and SD are more reasonable for a scale score built from several items. Many reports give both, which is fine as long as the reader can also see how the responses are distributed.