How is something distributed? (descriptive statistics)

In brief

One or more columns are to be described without testing anything: numbers with mean, standard deviation, smallest and largest value; groups with their frequencies. Columns with more than 12 distinct values count as numbers, the others as groups.

Requirements

None. Missing answers are reported and not included.

Reading the result

The mean describes the center, the standard deviation the spread. With skewed distributions or outliers the mean is of little use; the chart shows the shape. The figures describe only the people in these data.

Whether the values are normally distributed is shown by the Shapiro-Wilk test with the Q-Q plot; see "Is it normally distributed?". Histogram, box plot and bars can be chosen under "Charts".

The 5% trimmed mean (table "Robust measures of location") leaves out the lowest and highest 5% of the values, a boundary value counts in part, as in SPSS and PSPP. Small samples: SPSS cuts in part, R (mean with trim) cuts whole values. The values stay in the data. With fewer than 20 values it hardly protects against a single outlier.

For category columns "How diverse?" shows the IQV, Shannon H and the effective number of categories (see "How diverse?"); under "More options" there are bootstrap intervals for the mean and the median (see "Permutation test and bootstrap").

Comparing the mean with a fixed value (one-sample t test)

Below the result there is "Compare the mean with a fixed value": for each number column a comparison value can be entered that is fixed before the analysis (a norm value, the midpoint of the answer scale, a value from earlier studies). Statable calculates the one-sample t test as SPSS does: t = (M − comparison value) / (SD / √n) with n − 1 degrees of freedom, two-sided, plus the 95% confidence interval of the difference and Cohen's d with interval. The sentence and the line for the paper are in the result and in the report.

t(149) = 1.76, p = .080, d = 0.14, 95% CI [−0.02, 0.30] (comparison value 50)

Comparing a proportion with a fixed value (binomial test)

Below the result, category columns have "Compare a proportion with a fixed value": choose a category and enter a proportion that is fixed before the analysis (0.5 or 50%). Statable calculates the exact binomial test as in R (two-sided: all outcomes that are at most as likely as the observed one) with the Clopper-Pearson interval, plus the z test without correction like 1-PropZTest on the TI-84. If n · p₀ or n · (1 − p₀) is below 10, the exact value applies.

Comparing the spread with a fixed value (chi-square test for one variance)

For number columns there is also "Compare the spread with a fixed value": enter a standard deviation that is fixed before the analysis. Statable calculates χ² = (n − 1) · s² / σ₀² with n − 1 degrees of freedom, two-sided (the smaller tail doubled), and the 95% range of the standard deviation. The test is very sensitive to skewed values and outliers, even with many cases.

Typical examples

• Sleep duration of the respondents: mean, spread, histogram.
• Likert question (1 to 5): frequencies of each answer.
• Gender and school type: frequencies of the groups.

M = 6.80, SD = 1.42, n = 118