Descriptive statistics

• Descriptive statistics summarize and describe the main dataset features.
• They provide simple summaries about the sample and the measures.
• Descriptive statistics do not infer beyond the data, so they are not used to make conclusions regarding any hypotheses.
• They are typically distinguished from inferential statistics.

Types of Descriptive Statistics

• Measures of central tendency (mean, median, mode)
• Measures of dispersion or variability (range, variance, standard deviation)
• Measures of distribution shape (skewness, kurtosis)

Measures of Central Tendency

• Describe the typical or "center" value of a dataset.
• Common measures include mean, median, and mode.

Mean

• The average of all values in a dataset.
• Calculated by summing all values and dividing by the number of values.
• Sensitive to outliers.

Median

• The middle value in a dataset when the values are arranged in ascending or descending order.
• Divides the dataset into two equal halves.
• Less sensitive to outliers than the mean.

Mode

• The value that appears most frequently in a dataset.
• A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.

Measures of Dispersion

• Describe the spread or variability of values in a dataset.
• Common measures include range, variance, and standard deviation.

Range

• The difference between the maximum and minimum values in a dataset.
• Provides a simple measure of the total spread.
• Highly sensitive to outliers.

Variance

• The average of the squared differences from the mean.
• Measures how far each value in the dataset is from the mean.
• Larger variance indicates greater variability.

Standard Deviation

• The square root of the variance.
• Provides a more interpretable measure of variability than variance, as it is in the original units of the data.
• Commonly used to describe the spread of data around the mean.

Measures of Distribution Shape

Skewness

• Measures the asymmetry of a distribution.
• A symmetrical distribution has a skewness of 0.
• Positive skewness indicates a long tail extending towards higher values; the mean is typically greater than the median.
• Negative skewness indicates a long tail extending towards lower values; the mean is typically less than the median.

Kurtosis

• Measures the "tailedness" of a distribution.
• Indicates the degree to which scores cluster in the tails of the distribution and is a measure of "peakedness".
• High kurtosis indicates a distribution with heavy tails and a sharp peak.
• Low kurtosis indicates a distribution with light tails and a flat peak.

Graphical Descriptive Statistics

• Visual representations to summarize and present data.
• Include histograms, box plots, scatter plots, and more.

Histograms

• Display the frequency distribution of a continuous variable.
• Data is grouped into bins, and the height of each bar represents the frequency of values within that bin.
• Useful for visualizing the shape of the distribution (skewness, kurtosis) and identifying outliers.

Box Plots

• Display the median, quartiles, and outliers of a dataset.
• The box represents the interquartile range (IQR), which contains the middle 50% of the data.
• Whiskers extend from the box to the minimum and maximum values within a certain range (e.g., 1.5 times the IQR).
• Outliers are plotted as individual points beyond the whiskers.

Scatter Plots

• Display the relationship between two continuous variables.
• Each point on the plot represents a pair of values for the two variables.
• Useful for identifying patterns, trends, and correlations between the variables.

Descriptive Statistics for Categorical Data

• Focus on summarizing the frequency and proportion of different categories.
• Commonly use frequency tables, bar charts, and pie charts.

Frequency Tables

• Display the count and percentage of observations in each category.
• Provide a comprehensive summary of the distribution of a categorical variable.

Bar Charts

• Visually represent the frequency or proportion of each category.
• The height of each bar corresponds to the frequency or proportion of the category.
• Useful for comparing the relative sizes of different categories.

Pie Charts

• Display the proportion of each category as a slice of a circle.
• The size of each slice is proportional to the percentage of the category.
• Useful for visualizing the relative contribution of each category to the whole.

Descriptive Analysis Considerations

• The choice of descriptive statistics depends on the type of data and the research question.
• Understanding the properties and limitations of each statistic is crucial for accurate interpretation.
• Outliers can significantly affect some descriptive statistics, so it's important to identify and handle them appropriately.
• Visualizations enhance the understanding and communication of descriptive statistics.