Common mistake 1 with Scale reliability (Cronbach’s alpha)

Reading alpha as validity. A scale can be highly reliable and measure the wrong thing; alpha cannot tell you which construct you have.

Common mistake 2 with Scale reliability (Cronbach’s alpha)

Chasing a high alpha by adding near-duplicate questions. It rises mechanically with the number of items, so above about 0.95 the items are redundant rather than excellent - and respondents pay for every extra question.

Common mistake 3 with Scale reliability (Cronbach’s alpha)

Dropping items until alpha clears a threshold, then not saying so. That is fitting the scale to the sample, and the reliability will not replicate.

Common mistake 4 with Scale reliability (Cronbach’s alpha)

Reporting alpha for a scale that is then used one item at a time. The reliability of the total says nothing about the reliability of any single item, which is always lower.

EconStatsCraft › Scale reliability (Cronbach’s alpha)

Scale reliability (Cronbach’s alpha)

Whether a battery of survey questions hangs together as one measure. The first thing to run on any multi-item scale, and the number every marketing dissertation is asked for before the analysis that uses the scale.

What it needs

Scale items - The questions meant to measure ONE construct.

The intuition

Several survey items are meant to measure one underlying thing. Reliability asks whether they behave that way - whether people who score high on one score high on the others. Cronbach's alpha is the standard number, it is reported everywhere, and it is widely misunderstood: it is not a measure of unidimensionality, and it rises with the number of items whether or not those items are any good.

When to use it

Before any analysis that averages or sums several items into a construct - service quality, brand image, satisfaction. This is the number a marketing dissertation is asked for first, and the check that stops a meaningless composite being used as if it were a measurement.

Where the estimator comes from

For k items with total score X=\sum_j X_j, \alpha=\frac{k}{k-1}\left(1-\frac{\sum_j \sigma^2_{X_j}}{\sigma^2_X}\right). The bracket compares the sum of item variances against the variance of the total: if items are uncorrelated the total variance is just the sum and \alpha=0; if they are perfectly correlated the total variance is k^2 times larger and \alpha\to1. Written in terms of the average inter-item correlation \bar{r}, the Spearman-Brown form makes the dependence on length explicit: \alpha=\frac{k\bar{r}}{1+(k-1)\bar{r}}, which is increasing in k for any fixed \bar{r} - so length alone raises alpha.

The model

\alpha = \frac{k}{k-1}\left(1 - \frac{\sum_i \sigma^2_i}{\sigma^2_{\text{total}}}\right)

What it assumes

That the items are all scored in the same direction. A negatively worded item left as-is makes a perfectly reliable scale look broken - reverse-score it first.
That every item loads on the construct roughly equally (tau-equivalence). Where that is doubtful, alpha understates reliability and a factor analysis is the better tool.
One construct, not two. Alpha on a mixed bag can still look respectable, which is why the item-total correlations matter more than the headline.
That the respondents are drawn from a population with genuine variation in the construct. Alpha depends on between-person variance: a homogeneous sample can make a perfectly good scale look unreliable, and a heterogeneous one can rescue a poor one.

Statistical properties

Alpha rises with the number of items even when item quality is unchanged. That is arithmetic, not evidence, and it is why a long scale of mediocre items can post a respectable alpha.
Alpha is a LOWER BOUND on reliability, and only equals it under tau-equivalence - all items loading equally on the factor. When loadings differ it understates, and McDonald's omega is the better estimate.
A high alpha does not imply one dimension. Two distinct but correlated factors can produce alpha above 0.8, so factor analysis, not alpha, answers the dimensionality question.
The 0.7 threshold is a convention from Nunnally and was proposed for early-stage research, with 0.9 or above suggested for high-stakes decisions. It has hardened into a rule it was never meant to be.
Alpha above about 0.95 usually signals redundancy - items that are paraphrases of each other - which inflates reliability while adding no information.
Reverse-coded items must be reversed before computing alpha. Forgetting is the single commonest cause of an unexpectedly low value, and shows as a negative item-total correlation.

How to read the output

Around 0.7 is the usual floor for research use, 0.8 is comfortable. Below 0.6 the items are not measuring one thing.
A corrected item-total correlation below about 0.3 marks an item that does not belong. Corrected matters: correlating an item with a total that contains it is guaranteed to look good.
"Alpha if deleted" higher than the overall alpha is the actionable number - the scale is better without that item.
Alpha is a lower bound on reliability under its assumptions, not an estimate of it. Reporting it as 'the reliability' overstates precision, and where tau-equivalence is doubtful omega is the better statistic.

A worked reading

Four service-quality items on 600 respondents give alpha = 0.8812, comfortably above the conventional 0.7. The item analysis is what makes that number trustworthy rather than merely large: every corrected item-total correlation sits between 0.7196 and 0.7667, so all four items are measuring the same thing to a similar degree, and "alpha if deleted" is LOWER than 0.8812 for every one of them - dropping any item would make the scale worse, which is the pattern a coherent scale shows. Now the caution that alpha alone cannot give you. Alpha rises mechanically with the number of items: the same four items repeated to make twenty would push it well past 0.9 without measuring anything new. A high alpha on a long questionnaire tells you the questionnaire is long, and it is only evidence of reliability when read beside the item-total correlations, as here.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Alpha near zero or negative	A reverse-coded item not reversed	Check item-total correlations for a negative
Alpha rising sharply when an item is dropped	That item does not belong	Inspect its content, not just the statistic
Alpha above 0.95	Redundant items	Shorten the scale; nothing is lost
High alpha but two factors	Alpha does not test dimensionality	Run factor analysis
Alpha low with few items	Expected. Length drives it	Judge by average inter-item correlation instead
Alpha differing from omega	Loadings are unequal	Report omega; alpha is a lower bound

Common mistakes

Reading alpha as validity. A scale can be highly reliable and measure the wrong thing; alpha cannot tell you which construct you have.
Chasing a high alpha by adding near-duplicate questions. It rises mechanically with the number of items, so above about 0.95 the items are redundant rather than excellent - and respondents pay for every extra question.
Dropping items until alpha clears a threshold, then not saying so. That is fitting the scale to the sample, and the reliability will not replicate.
Reporting alpha for a scale that is then used one item at a time. The reliability of the total says nothing about the reliability of any single item, which is always lower.

Where to read this properly

Cronbach (1951), "Coefficient Alpha and the Internal Structure of Tests", Psychometrika 16(3).
Sijtsma (2009), "On the Use, the Misuse, and the Very Limited Usefulness of Cronbach's Alpha", Psychometrika 74(1).
McDonald, Test Theory: A Unified Treatment (1999), for omega.
Nunnally, Psychometric Theory (1978), the source of the 0.7 convention.

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Scale reliability (Cronbach’s alpha)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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