# 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.

## Z-Scores

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.

## Correlation

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

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

## Visualize Your Data with Box and Whisker Plots!

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