Descriptive Statistics Cheat Sheet

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Types of Data

Nominal: Categorical data without an inherent order.
Ordinal: Categorical data with a defined order but not evenly spaced.
Interval: Numerical data with equal intervals but no true zero.
Ratio: Numerical data with equal intervals and a true zero.

Measures of Central Tendency

Provide a central value for the data set.

Mean (Average): \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\)
Median: Middle value when data is ordered.
Mode: Most frequently occurring value.

Measures of Dispersion

Indicate the spread or variability of a data set.

Range: Difference between the highest and lowest values.
Variance: Average of the squared differences from the Mean. \(\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}\) for a population, \(s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}\) for a sample.
Standard Deviation: Square root of the variance. \(\sigma\) for population, \(s\) for sample.
Interquartile Range (IQR): Difference between the 75th percentile (Q3) and the 25th percentile (Q1).

Skewness and Kurtosis

Skewness: Measure of the asymmetry of the probability distribution.
Kurtosis: Measure of the 'tailedness' of the probability distribution.

Graphical Representations

Visualize data to identify patterns, trends, and outliers.

  • Bar Chart: Represents categorical data with rectangular bars.
  • Histogram: Represents the distribution of numerical data.
  • Box Plot: Visual representation of the five-number summary (Minimum, Q1, Median, Q3, Maximum).
  • Scatter Plot: Shows the relationship between two quantitative variables.


Measure of how many standard deviations an element is from the mean.

\(z = \frac{x - \bar{x}}{\sigma}\) where \(x\) is a score from the population, \(\bar{x}\) is the mean of the population, and \(\sigma\) is the standard deviation of the population.


Measure of the strength and direction of a linear relationship between two variables.

\(r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}\)

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