Measuring Spread with Standard Deviation

Categorical Variables

Variables dividing cases into distinct groups.

Quantitative Variable

Variable measuring numerical quantity for each case.

Population

All individuals or objects of interest.

Sample

Subset of the population used for inference.

Descriptive Statistics

Numerical/graphical methods to summarize data patterns.

Data Distribution

Tells values a variable takes and their frequencies.

Frequency Table

Shows number of cases in each category.

Relative Frequency

Proportion of cases in a category.

Proportion

Ratio of category instances to total observations.

Bar Chart

Graphical representation of categorical variable frequencies.

Bar Chart Titling

Guidelines for labeling and titling bar charts.

Pie Chart

Circular chart divided into sectors to show proportions

Pareto Chart

Bar chart with bars in decreasing order of frequency

Two-Way Table

Table showing relationship between two categorical variables

Mode

Most frequently occurring category in a distribution

Unimodal

Distribution with one distinct mode

Bimodal

Distribution with two modes of similar frequency

Multimodal

Distribution with more than two modes of similar frequency

Variability

Diversity of categories in a categorical distribution

Interpretation of Distribution

Analyzing the values and frequencies of a variable

Two Categorical Variable Relationship

Investigating the association between two categorical variables

Proportion Calculation

Determining the ratio or percentage of a subset in a sample

Difference in Proportions

Calculation of the variance in proportions for different categories

Segmented Bar Chart

Bar chart with segments representing different categories

Side-by-Side Bar Chart

Separate bar charts for each group of a categorical variable

Distribution of a Variable

Describes what values the variable takes and how often it takes these values.

Dot Plot

Records data values on a number line with a dot for each observed value, showing frequency and variation.

Outliers

Values notably distinct from other values in a dataset, often much larger or smaller.

Histogram

A graph for quantitative data that groups values into intervals (bins) showing frequency.

Bin Width

The difference between consecutive lower class limits in a histogram.

Number of Bars in Bar Chart

Equals the number of categories.

Number of Bars in Histogram

Varies and can be determined by the user or software.

Bar Order in Bar Chart

Order does not matter as it graphs categorical data.

Bar Order in Histogram

Must be presented in numerical order.

Width of Bars in Bar Chart

Meaningless and determined arbitrarily by software.

Width of Bars in Histogram

Defined as the bin width and can be edited.

Gaps between Bars in Bar Chart

Indicate impossibility of observations between categories.

Gaps between Bars in Histogram

Indicate no values were observed in the bin or class.

Shape of Distribution

Describes if the distribution is symmetric, mound-shaped, or has peaks or clusters.

Center of Distribution

Indicates where the distribution is centered and the typical value.

Variability of Data

Describes how spread out the data is and if most values are within a certain range.

Unusual Observations

Identifies outliers that deviate markedly from the overall pattern.

Smooth Curve in Histogram

Illustrates the general shape of the distribution with less jagged edges.

Interpreting Histogram

Involves analyzing the shape, center, variability, and unusual observations of the distribution.

Characteristics of Shape

Include symmetry, number of peaks, and presence of unusually large or small values.

Common Shapes of Distributions

Include symmetric shapes like bell-shaped and asymmetric shapes like right or left skewed.

Symmetric Shape: Bell-Shaped

Distribution where values fall in the middle, frequencies tail off symmetrically, and left and right halves mirror each other.

Symmetric Shape: Uniform

Distribution where bars tend to occur with similar frequency, creating a flat histogram.

Skewed Shape: Right-Skewed

Distribution where the tail extends to the right, with more data in the lower end.

Skewed Shape: Left-Skewed

Distribution where the tail extends to the left, with more data in the upper end.

Shape: Peaks

Classifies data by the number of peaks or mounds present (unimodal, bimodal, or multimodal).

Notes about Peaks or Mounds

Peaks of different heights in distributions may indicate distinct groups within the data.

Numerical Measures of Center and Spread

Focus on shape, center, and spread of the distribution for precise interpretation.

Populations and Samples

Populations include all individuals, while samples are portions; parameters for populations and statistics for samples.

Numerical Measures of Center

Values representing the average or typical value of a quantitative variable.

Measure of Center: Mean

Arithmetic mean, the average of data set items, calculated by summing values and dividing by the number of values.

Summation Notation

Used to sum data set observations together, identified by subscripts.

Mean Formulas

Population mean (m) and sample mean (x-bar) formulas for center measures.

Sample Mean

Uses the symbol 𝑥 (read x-bar) and is a statistic calculated using statistical software for a data set.

Median

The middle value or the value that splits the data in half; denoted as m.

Calculating Median

Ordering quantitative data from smallest to largest and determining the median's location using the formula 𝐿(m) = (𝑛+1)/2.

Measure of Center

The mean is sensitive to outliers and skewed distributions, while the median is resistant to outliers.

Comparing Mean and Median

Both provide measures of center, with the mean affected by outliers and skewness, unlike the median.

What to Do with Outliers

Consider if outliers are errors or part of the population; run analyses with and without outliers to assess their impact.

Measuring Spread

Variability in a quantitative distribution measured by how far data points spread along the x-axis.

Understanding Variation

Variation exists when data values differ from the mean, indicating the spread in the data set.

Standard Deviation

Measures how far observations are from the mean, with most data falling within one standard deviation.

Standard Deviation Calculation

Involves finding the mean, calculating deviations from the mean, squaring each deviation, summing them, and taking the square root to get the standard deviation.