Types of Data in Statistics

Types of Data

• Qualitative (categorical) data: • Measured by classification • Non-numerical in nature • Categories with a meaningful order are ordinal data • Categories without a meaningful order are nominal data • Typically, categories are mutually exclusive and collectively exhaustive
• Quantitative (numerical) data: • Measured on a naturally occurring scale • Allow for meaningful mathematical calculations

Cross-Sectional and Time Series Data

• Cross-sectional data: • Collected at the same or approximately the same point in time • Example: average age of people in each state in 2013
• Time series data: • Collected over several consecutive time periods for the same unit • Example: a company's daily stock price in April 2014
• Panel data: • Combines both cross-sectional and time series data

Key Terms

• Population (universe): • Data on all units/items of interest
• Sample: • Portion of population
• Parameter: • Summary measure about population
• Statistic: • Summary measure about sample

Samples

• Samples need to be: • Representative to reflect the population of interest • Random to ensure that each subset of fixed size is equally likely to be selected • Large, the more data the better

Presenting Qualitative Data

• Frequency table: • A summary of data showing the frequency (or count) of items in each of several non-overlapping classes • Relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class • Percent frequency of a class is the relative frequency multiplied by 100
• Bar graph: • Bars (or columns) arranged in descending order of the values from top to bottom (or from left to right)
• Pie chart: • A circular graph divided into sectors to show proportion of each category
• Pareto diagram: • Bars (or columns) arranged in descending order of the values from top to bottom (or from left to right)

Presenting Quantitative Data

• Frequency distribution: • Determine range • Select the number of classes (usually between 5 and 20) • Compute class intervals (width) • Determine class boundaries (limits) • Count observations in each class
• Histogram: • A common graphical presentation of quantitative data • The variable of interest is placed on the horizontal axis, and the frequency, relative frequency, or percent frequency is placed on the vertical axis • A rectangle is drawn above each class interval with its height corresponding to the interval's frequency
• Dot plot: • A simple graphical summary of data • A horizontal axis shows the range of data values • Each data value is represented by a dot placed above the axis
• Stem-and-leaf display: • Shows both the order of data and the shape of the distribution of the data • Similar to a histogram, but it has the advantage of showing the actual data values • Divide each observation into stem value and leaf value • Stem value defines the class • Leaf value defines the frequency (count)

Graphing Bivariate Relationships

• Describes a relationship between two variables
• Plotted as a scatterplot/scattergram

Measures of Variability

• Variability: • The spread of the data across possible values • Commonly used measures of variability: range, interquartile range, variance, and standard deviation
• Range: • Difference between the largest and the smallest values • Ignores how data are distributed • Sensitive to outliers
• Quartiles: • Split ordered data into 4 segments (4 quarters) with an equal number of values in each segment • Position of i-th quartile: [i(n+4)]/4
• Interquartile range: • Also called midspread • Difference between the third and first quartiles (spread in the middle 50%) • Not affected by extreme values
• Sample variance: • The sum of the squared deviations from the mean divided by (n-1) • Expressed as "units" squared
• Sample standard deviation: • The positive square root of the sample variance • Most commonly used measure of variation • Shows variation about the mean • Expressed in the original units of the data

Interpreting the Standard Deviation

• Chebyshev's rule: • A rule of thumb that applies to any set of data regardless of the shape of the distribution • In general, for k > 1, at least 1 - 1/k^2 of the data lies within the interval (x̄ - k * s, x̄ + k * s)
• Empirical rule: • A rule of thumb that applies to mound-shaped and symmetric distributions only • Approximately 68% of the data lies within 1 standard deviation of the mean • Approximately 95% of the data lies within 2 standard deviations of the mean • Approximately 99.7% of the data lies within 3 standard deviations of the mean

Using Descriptive Statistics

• Begin with a discussion of the "center" of the data, generally based on the mean
• Follow with a discussion of variability (and skew if appropriate)
• End with a summary evaluation that may have a subjective component
• Make sure to use numbers in a description wisely – not too few or too many