Descriptive Statistics: Measures of Central Tendency and Variability

Descriptive Statistics

Measures of Central Tendency

• Mean: The average value of a dataset, calculated by summing all values and dividing by the number of values.
• Median: The middle value in a dataset when it is arranged in order, used when the dataset contains outliers.
• Mode: The most frequently occurring value in a dataset.

Measures of Variability

• Range: The difference between the largest and smallest values in a dataset.
• Variance: The average of the squared differences between each value and the mean.
• Standard Deviation: The square root of the variance, used to measure the spread of a dataset.

Inferential Statistics

Hypothesis Testing

• Null Hypothesis (H0): A statement of no effect or no difference.
• Alternative Hypothesis (H1): A statement of an effect or difference.
• p-value: The probability of observing the result by chance, used to determine whether to reject the null hypothesis.

Confidence Intervals

• Confidence Level: The probability that the interval contains the population parameter (e.g., 95%).
• Margin of Error: The maximum amount by which the sample statistic may differ from the population parameter.

Data Visualization

Types of Plots

• Histogram: A graphical representation of a dataset, showing the frequency of each interval of values.
• Scatter Plot: A graphical representation of the relationship between two variables.
• Bar Chart: A graphical representation of categorical data, showing the frequency of each category.

Common Statistical Distributions

Discrete Distributions

• Bernoulli Distribution: Models the probability of a binary outcome (e.g., success or failure).
• Binomial Distribution: Models the probability of k successes in n trials.

Continuous Distributions

• Normal Distribution: A continuous distribution with a symmetric, bell-shaped curve.
• Uniform Distribution: A continuous distribution with equal probability across a fixed interval.

Descriptive Statistics

Measures of Central Tendency

• The mean is sensitive to outliers and is used when the dataset is normally distributed or symmetric.
• The median is a better representation of central tendency when the dataset contains outliers or is skewed.
• The mode is used when the dataset has multiple modes or peaks.

Measures of Variability

• The range is affected by outliers and is not resistant to extreme values.
• Variance is sensitive to extreme values and outliers, making it a less robust measure.
• Standard deviation is used to compare the spread of different datasets.

Inferential Statistics

Hypothesis Testing

• A null hypothesis proposes no effect or no difference, while an alternative hypothesis proposes an effect or difference.
• The p-value determines the probability of observing the result by chance, and a low p-value (usually < 0.05) indicates rejection of the null hypothesis.

Confidence Intervals

• The confidence level determines the probability that the interval contains the population parameter.
• The margin of error determines the maximum amount by which the sample statistic may differ from the population parameter.

Data Visualization

Types of Plots

• Histograms are used to visualize the distribution of continuous data.
• Scatter plots visualize the relationship between two continuous variables.
• Bar charts visualize categorical data and show the frequency of each category.

Common Statistical Distributions

Discrete Distributions

• Bernoulli distribution models binary outcomes, such as success or failure.
• Binomial distribution models the number of successes in a fixed number of trials.

Continuous Distributions

• Normal distribution is a continuous, symmetric, and bell-shaped distribution used to model real-valued variables.
• Uniform distribution is a continuous distribution with equal probability across a fixed interval, used to model random variables with equal likelihood.