Statistical note

Descriptive and robust statistics with SciPy

Mean, median, variance, IQR, MAD, skewness, and kurtosis: what each summary measures and when it can mislead.

18 July 20269 min read

Summary statistics compress a distribution, but each compression preserves different information. For observations $x_1,\ldots,x_n$ , the arithmetic mean is

$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i,$

while the sample variance is

$s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2.$

Mean, median, IQR, and outliers on a skewed distribution

Robust alternatives

The median resists isolated extreme values. The interquartile range is $IQR=Q_{0.75}-Q_{0.25}$ . The median absolute deviation is $MAD=\operatorname{median}(|x_i-\operatorname{median}(x)|)$ ; multiplying by approximately $1.4826$ makes it comparable to the standard deviation under normality.

import numpy as np
from scipy import stats

area = np.array([42, 45, 46, 48, 51, 53, 55, 58, 190])
summary = stats.describe(area)
print(summary.mean, np.median(area))
print(stats.iqr(area), stats.median_abs_deviation(area, scale="normal"))
print(stats.skew(area, bias=False), stats.kurtosis(area, bias=False))

Case study: segmented cell area

One merged segmentation can make the mean cell area jump while the median changes little. Report a distribution plot and robust summaries before interpreting a treatment effect. Skewness describes asymmetry; kurtosis describes tail weight relative to a reference convention, not simply “peakedness.”

Functions: scipy.stats.describe, iqr, median_abs_deviation, skew, and kurtosis.

Robust alternatives

Case study: segmented cell area

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