Cronbach’s alpha calculator

What does Cronbach’s alpha tell you?

Cronbach’s alpha is a number, usually between 0 and 1, that summarizes how consistently the items on a questionnaire or test move together. If a ten-item anxiety scale is really measuring one thing, a person who scores high on one item should tend to score high on the others, and alpha will be high. The formula, introduced by Lee Cronbach in 1951, is α = k ÷ (k − 1) × (1 − Σσᵢ² ÷ σₓ²), where k is the number of items, σᵢ² is the variance of item i across respondents and σₓ² is the variance of the respondents’ total scores. When items are unrelated the total variance is no bigger than the sum of the item variances and alpha is near 0; when they move in lockstep the total variance is far larger and alpha approaches 1. It is a statement about a particular sample taking a particular test, not a permanent property of the questionnaire.

Worked example: six respondents, four items

Take six people answering four items on a 1–5 scale (rows: 4,5,4,3 · 3,4,4,3 · 5,5,4,5 · 2,3,2,2 · 4,4,5,4 · 3,2,3,3). The four item variances add up to 4.60, and the variance of the six total scores (16, 14, 19, 9, 17, 11) is 14.27. Then α = 4 ÷ 3 × (1 − 4.60 ÷ 14.27) = 0.903. The mean correlation between the items is 0.707, so the standardized alpha is 4 × 0.707 ÷ (1 + 3 × 0.707) = 0.906. The first item correlates 0.92 with the sum of the other three, and dropping it would lower alpha to 0.824; dropping item 2 would raise it slightly to 0.916. With only six respondents this is an illustration of the arithmetic, not evidence about the scale. A second check from the textbook-style matrices on Wikipedia: six items with variance 10 and covariance 3 give a total variance of 150 and α = 0.72; with covariance 1 the total variance is 90 and α = 0.40.

What counts as a good alpha, and why is 0.90+ not always better?

Tavakol and Dennick (2011) note that reports of acceptable values range from 0.70 to 0.95, and that a maximum of 0.90 has been recommended: an alpha above 0.90 can mean some items are redundant, so the test could be shortened without losing information. A common rule of thumb treats 0.70 as acceptable for research on groups, but the right bar depends on the stakes. Scores used to make decisions about individuals need much higher reliability than group averages used in a survey. There is no universal threshold; this calculator only labels the value against those published ranges and says so.

Does a high alpha prove the scale measures one thing?

No. Alpha rises with the number of items and with the average correlation, and it does not test dimensionality. Wikipedia gives a numerical illustration: a uni-dimensional six-item matrix and a two-factor matrix can produce exactly the same alpha of 0.72. Tavakol and Dennick make the same point, that a multidimensional test does not necessarily have a lower alpha than a unidimensional one, and advise calculating alpha separately for each construct in a multi-construct instrument. Check dimensionality first, with factor analysis or at least the item correlation pattern, then report alpha for each scale or subscale.

What is the Spearman–Brown formula and how do I use it here?

The Spearman–Brown prediction formula says how reliability changes if you lengthen or shorten a test with items of the same quality: ρ* = nρ ÷ (1 + (n − 1)ρ), where ρ is the current reliability, n is the factor by which length changes and ρ* is the predicted reliability. Doubling a test with reliability 0.70 predicts 0.82. Solving for n gives the length needed for a target: n = ρ*(1 − ρ) ÷ (ρ(1 − ρ*)). Raising a reliability of 0.80 to 0.95 needs n = 4.75, so a 50-item test would need about 238 comparable items. In the calculator, fill in the optional boxes under the data; if you leave the starting reliability blank, the alpha just calculated is used. The formula assumes the new items are parallel to the old ones, which is rarely exactly true, so treat the answer as a planning estimate.

What is “alpha if item deleted”, and should I remove items because of it?

It recomputes alpha as if one item had never been asked, and the “item-total r” column shows how strongly each item correlates with the sum of the others. A negative or near-zero item-total correlation often points to a reverse-keyed item that was not recoded, so fix the coding before you judge the item. Wikipedia warns that deleting items based on alpha-if-deleted can inflate alpha in the sample and may lower reliability in the population, recommends removing items on theoretical grounds and cross-validating on a split sample. Use the column to find items worth reading again, not as an automatic pruning rule.

How many respondents do I need?

The sources we checked give no single rule, and alpha is a sample-specific estimate: Tavakol and Dennick advise recomputing it every time a test is administered instead of quoting a published value. In practice the estimate is noisy with a few dozen respondents, and the confidence interval on alpha can be wide, so report the sample size alongside the value. Missing answers must be handled before the calculation; this tool requires a complete table (every respondent answers every item).

Limits of this method

Alpha is an accurate reliability estimate only if the items are essentially tau-equivalent, meaning they measure the same thing on the same scale with independent errors; otherwise it understates reliability. It can also overstate it: rater bias, collusion or faking raise the covariance between items without raising true reliability. It is not a measure of validity, since a scale can be perfectly consistent and still measure the wrong thing. And it says nothing about stability over time, which needs a test–retest design. The “internal consistency” label is itself vaguely defined.

For general information only. Sources: L. J. Cronbach, “Coefficient alpha and the internal structure of tests,” Psychometrika 16(3) (1951) 297–334 (the original was not opened; the formula is taken from the encyclopedia article below); M. Tavakol and R. Dennick, “Making sense of Cronbach’s alpha,” International Journal of Medical Education 2 (2011) 53–55, doi:10.5116/ijme.4dfb.8dfd; Wikipedia, “Cronbach’s alpha” and “Spearman–Brown prediction formula” (original papers: C. Spearman, 1910, and W. Brown, 1910, British Journal of Psychology 3, not opened). The standardized-alpha formula is the standard one and is not printed in the Tavakol and Dennick paper. Related: Cohen’s kappa calculator, correlation coefficient calculator and standard deviation calculator. How we check our calculators.