IEE 380: Numerical Data Summaries

• Descriptive statistics is the focus
• The material is from IEE 380: Probability and Statistics for Engineering Problem Solving

Numerical Summaries of Data

• Central tendency measures include mean, median, and mode

• Mean is the average of the data points

• Median is the middle value when data is ordered from smallest to largest; for an even number of values, average the two middle values

• Mode is the most frequently occurring value(s)

• Variability, Dispersion, or Spread measures include range, variance, standard deviation, and interquartile range (IQR)

• Range is the difference between the maximum and minimum values

• Variance is the average of the squared differences between each data point and the mean

• Standard deviation is the square root of the variance, with the same units as the mean

• Interquartile Range (IQR) is the difference between the 75th percentile (Q3) and the 25th percentile (Q1), measuring the spread of the middle 50% of the data

Shape of the Distribution

• Skewness measures the asymmetry of the distribution
• Positive skew indicates a longer right tail
• Negative skew indicates a longer left tail
• Kurtosis measures the thickness of the tails of the distribution:
• High kurtosis means heavier tails
• Low kurtosis means lighter tails

Percentiles and Quartiles

• Percentiles indicate the value below which a certain percentage of the data falls
• Quartiles divide the data set into four equal parts
• Q1 = 25th percentile
• Q2 = 50th percentile
• Q3 = 75th percentile

Other Numerical Summaries

• Five-Number Summary includes the minimum, Q1, median, Q3, and maximum
• Coefficient of Variation (CV) is the standard deviation divided by the mean and is used for comparing variability across datasets with different units
• Z-scores indicate the number of standard deviations a data point is from the mean

Sample Mean

• The sample mean measures central tendency
• The sample mean differs from the population mean, denoted by versus
• Sample Mean Example: Pull-off Forces, computes the average of engine connector pull-off force observations

Sample Variance

• Sample variance measures spread or dispersion of a dataset and differs from the population variance, denoted versus
• Sample Variance Example: Pull-off Forces, uses calculator functions like Stat Edit, entering data into a list (e.g., L1), Stat Calc 1-VarStats, and produces and
• Adjustment is made to estimate population variance and called “degrees of freedom"

Frequency Distributions

• Frequency distributions are compact summaries of data, expressed as a table, graph, or function
• Components include categories or intervals (classes/bins) and frequency
• For categorical data, categories are utilized for summary
• For numerical data, intervals are utilized (e.g., 1-10, 11-20, etc.)
• Frequency is the number of data points in each category or interval
• Absolute frequency is the number of data points in each category or interval
• Relative frequency is the proportion of the total data points in each category or interval

Visualization of Frequency Distributions

• Common visualizations include bar charts, histograms, and frequency polygons
• Bar Charts are for categorical data, often displayed as a Pareto Chart
• Histograms are for numerical data grouped into intervals
• Frequency Polygons are line graphs connecting the midpoints of intervals
• Visualization helps make the data easier to interpret
• Visualization compares between different datasets
• Visualization as a foundation for statistical analyses

Building Histograms

• Steps:
• Collect and organize numeric data, sorting in ascending order is helpful
• Determine the range (Maximum value – Minimum value)
• Decide on the number of bins using the square root of the number of data points or other methods
• Calculate the Bin Width (Range/Number of bins)
• Define Bin Intervals: starting from the minimum value and adding the bin width repeatedly to create intervals, ensuring intervals do not overlap
• Count Frequencies for each bin and tally how many data points fall within each bin
• Draw the histogram: The x-axis represents the bin intervals
• The y-axis represents the frequencies
• Draw the width of each bar equal to the bin width
• The height of each bar is equal to the frequency
• Label appropriately

Histogram Example: Compressive Strength

• Example includes data
• Range, number of bins, bin width, starting point (70), ending point (250) along with tallying the frequencies are all considerations to creating this visualization

Poor Choices in Drawing Histograms

• Too many bins can create a jagged shape and result in errors
• When the horizontal axis scale is not at class boundaries that creates errors
• When horizontal axis labels does not include units, that introduces errors

Shapes of Frequency Distributions

• Shapes indicate Negative (left) skew
• Shapes indicate Symmetrical distribution
• Shapes indicate Positive (right) skew

Pareto Charts

• Used to represent frequency distributions for categorical data