Types of Data in Statistics

Study Notes

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 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

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

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)

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)

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

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

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