Teaching · For Teachers · Research and data

The shape of your data,
in two numbers
and a picture.

Before averaging scores or running a t-test, check the shape. Paste a list and the calculator gives skewness and kurtosis with their standard errors, draws the histogram, and says in plain words what the shape means for the analysis you were about to run.

Check the shapeWhat you get
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Included with Caveat Classroom and Whole-school licenses. Or get it in the For Teachers toolkit, $79 for 72 items.

The skewness and kurtosis calculator

The worked example is made up: days absent in one homeroom. Use codes, not names. Nothing you type leaves this browser.

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What you get

Teaching · For Teachers$12 · included with Classroom and Whole-school licenses

What you finish with
Sample and population skewness and excess kurtosis, standard errors and z ratios, mean, median and SD, a histogram, and a plain reading of the shape. It prints as a summary and downloads as a spreadsheet.
Who it is for
Teachers and data teams checking a set of scores, school psychologists, education researchers and graduate students in education.
Format
Browser calculator for a phone, tablet or computer, by touch or keyboard. Four to 5,000 scores. Prints a summary; downloads a spreadsheet (.csv); export and import a file.
Access
$12 one-time, yours on every device you sign in on; the tool opens once you are signed in with it. Included with Caveat Classroom and Whole-school licenses. Also in the For Teachers toolkit ($79) and the Teaching Suite pass ($199), and 1 more bundle. Everything you type stays in this browser; nothing is sent to Caveat.
Where it lives
The For Teachers space in Caveat Teaching.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Say what the scores are.
  2. Paste the scores.
  3. Read the shape description and the histogram.
  4. Compare the z ratios with ±1.96 for a small sample, or the values themselves for a large one.
  5. Print the summary or download the spreadsheet.

Check the shape

How the numbers work

Moments about the mean, adjusted for sample size.

The worked example: days absentCentral moments24 scoresm₂, m₃, m₄ about the meanPopulation g11.678m₃ ÷ m₂^1.5Sample G11.792× √(n(n − 1)) ÷ (n − 2)z ratio3.79G1 ÷ SE 0.472The worked example: days absentCentral moments24 scoresm₂, m₃, m₄ about the meanPopulation g11.678m₃ ÷ m₂^1.5Sample G11.792× √(n(n − 1)) ÷ (n − 2)z ratio3.79G1 ÷ SE 0.472
Skewness averages the cubed distances from the mean, so one student with 16 days absent pulls it far more than several with 1 or 2. The sample formula then corrects for the small class.

Which skewness

There are three common formulas. g1 and g2 describe the scores you have. G1 and G2 adjust them so they estimate the population without bias for a normal curve; they are what SPSS, SAS and Excel’s SKEW and KURT report, so they are the headline here. All kurtosis on this page is excess kurtosis: a normal curve scores 0, not 3.

Reading the result

For small samples, divide each statistic by its standard error; beyond about ±1.96 the shape departs from normal by more than chance. In samples of several hundred, the standard errors shrink so far that trivial departures look significant, so judge the values: skewness beyond ±2 or kurtosis beyond ±7 is substantial. Counts such as absences or referrals are almost always skewed right, so lead with the median.

Sources

  1. Joanes, D. N., and Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society, Series D (The Statistician), 47(1), 183–189.
  2. Kim, H.-Y. (2013). Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis. Restorative Dentistry and Endodontics, 38(1), 52–54.
  3. NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.11, Measures of Skewness and Kurtosis (itl.nist.gov).
  4. Sturges, H. A. (1926). The choice of a class interval. Journal of the American Statistical Association, 21(153), 65–66.

The worked example is made up. Use student codes, not names. Your scores stay in this browser. Support: team@caveatacademic.com.