A questionnaire claims to measure ONE thing with many items. Cronbach’s α asks whether the items actually pull together — internal consistency. High when items correlate, near zero when they’re unrelated. Slide the items from independent to consistent.
Cronbach’s α = (k/(k−1))(1 − Σσᵢ²/σₜ²): with k items, it compares the summed item variances to the variance of the total score. When items correlate the total variance grows faster than the item variances, pushing α toward 1; independent items leave α near 0. The standard internal-consistency index (≥0.70 conventionally acceptable). A fail-loud self-check throws unless correlated items give α>0.70 within [0,1]. ◆ real statistics, node-verified.
Synthetic item scores are the illustration; the variance-ratio is the exact α. High α also just rises with MORE items and can mask multidimensionality — a known limit a careful peer names, not a proof of unidimensionality.