Descriptive Data Measures

• This material covers descriptive measures for data summarization.
• Data is summarized using measures of central tendency.
• Measures of variation are used to describe data.
• The position of a data value in a set is identified using measures of position.
• Techniques of exploratory data analysis are used.
• Stem and leaf plots, box plots, and five-number summaries enable discovery.

Measures of Central Tendency

• Two types of means are computed: one for a sample and one for a finite population.
• The sample mean uses the symbol X.
• The sample mean formula is X = (X₁ + X₂ + ... + Xₙ) / n = ΣX / n.
• The population mean uses the Greek symbol μ, pronounced "mu".
• The population mean is calculated as μ = (X₁ + X₂ + ... + Xₙ) / N = ΣX / N.
• N represents the size of the finite population.
• The mean may not be representative of the data in some situations.
• One extreme value can pull the mean upward.
• Every interval and ratio level dataset has a mean.
• All data values are included in the calculation of the mean.
• A dataset has one unique mean.
• The mean is useful for comparing two or more populations.
• The sum of deviations of each value from the mean is always zero.
• The mean is highly affected by extreme data.
• The median splits ordered data into halves.
• The symbol used to denote the median is mₑ.
• To find the median, arrange data in order and select the middle point.
• With an even number of values, the median is the average of the two middle numbers.
• The median grade for ordinal data can be determined
• A set of data has only one median.
• The median is not influenced by extremely large or small values.
• The median can be computed for ratio, interval, and ordinal-level data.
• Fifty percent of observations are greater and fifty percent are less than the median.
• The mode is the score that occurs most frequently denoted by M.
• The mode can be found for all levels of data.
• The mode is not affected by extremely high or low values.
• A dataset can have more than one mode; two modes indicates bimodal data.
• A disadvantage the set of data may not have a mode because no value appears more than once
• The weighted mean used when values in a data set are not all equally represented.
• The weighted mean of a variable X is calculated by weighting each value and dividing by the sum of the weights.
• Xw= (w₁X₁ + w₂X₂ + ... + wₙXₙ) / (w₁ + w₂ + ... + wₙ) = ΣwX / Σw, where w₁, w₂, ..., wₙ are weights.
• For nominal variables, the best measure of central tendency is the mode.
• For ordinal variables, the best measure is the median.
• For interval/ratio data, use the mean if not skewed and the median if skewed.
• In a symmetric distribution, the mean equals the median, equals the mode.
• With data skewed left, the mean is usually smaller than the median.
• With data skewed right, the mean is usually larger than the median.

Measures of Dispersion (Variation)

• Measures the spread or variability in a dataset
• Tells how meaningful measures of central tendency are
• Helps identify outliers or extreme scores
• Range of a variable is the difference between the largest and smallest.
• R = highest value – lowest value
• Only two values are used in the calculation of the range.
• Range is influenced by extreme values.
• The range is easy to compute and understand.
• The variance is based on the deviation from the mean. Calculate these deviations
• (xi – μ ) for populations
• ( xi –x ) for samples
• Deviations are squared ( x₁ – μ )² and (x; - )² for populations and samples respectively
• Population variance is sum of squared deviations from the mean, divided by population size.
• Represented by squared.
• Standard deviation (σ) is the square root of the variance.
• Small values mean scores are clustered mean, large values scattered.
• Influenced by extreme scores
• Units are squared original units.
• All values are used in the calculation.
• Always greater than or equal to zero, equal to zero only if all observations are the same
• Sample variance is the sum of squared deviations, divided by one less than the sample size..
• Uses n-1 degrees of freedom.

Range Rule of Thumb for Identifying Significant Values

• Significantly low values are μ – 2σ or lower.
• Significantly high values are μ + 2σ or higher.
• Values not significant are between (μ – 2σ) and (μ + 2σ).
• The standard deviation is used to measure the spread of the data.
• Small standard deviation indicates data clustered close to the mean.
• A large standard deviation indicates data spread out from the mean.
• Coefficient of Variation (CV) is a relative measure of standard deviation, as a percentage.
• CV = (σ/μ) * 100% or CV = (s/x) * 100%
• The coefficient of variation should only be computed for data on a ratio scale.
• The coefficient of variation is useful because it is unitless.
• When comparing datasets use the measures of variation instead of standard deviation.
• When the mean is near zero, the CV is sensitive to changes, limiting usefulness.

Measure of Position

• Quartiles divide data into four equal parts.
• Procedure is demonstrated in "Example For the following data set: 2, 3, 5, 6, 8, 10, 12 Find Q1 and Q3"
• Percentiles (P₁, P₂, ..., P₉₉) divide data into 100 groups, each with 1% of the values.
• Ogives visually represent cumulative frequency distributions.
• The interquartile Range (IQR) is Q₃ – Q₁.
• The interquartile range isis also called the midspread, middle fifty or inner 50% data range
• Outliers are extremely high or low data values.
• A data value is compared to lower and upper outlier fences to determine the range.
• Data points are considered an outlier where
  • The lower face has the forumla: X < lower fence
  • The upper face has the forumla: X > upper fence ,
• Dispersion describes data set variance .
• Skewness describes its directional variance.
• Skewness measures symmetry lack.
• Pearson’s coefficient of Skewness measures distribution symmetry degree and direction.
• sk₂ = 3(mean - median) / s

EDA: Exploratory Data Analysis

• Box and Whisker Plots graphically show a 5-number summary.
• Minimum value (excluding outliers) is smallest.
• The Minimum Whister begins at
• The first quartile Whister (Q1) begins after that
• The median Q2 is displayed next
• The third quartile Q3 comes before
• Finally the maximum value (excluding outliers) is at the end
• EDA is useful when sets is small or histograms do not work
• Collect data, arrange from lowest to highest, then find quartile difference.
• Obtain max and min values then label axes.
• Box plots show subgroup location and variation, and identify outliers.
• Box plots with the median near the center, similar length
• The medians position and length are critical to understand data skewness -The median is in the middle, and the whiskers are about the same on both sides -Median is closer to the bottom -Skewed to the left
• Plot interpretation using the data box plot : -It must be near the center to be consider symmetric -If the median must fall right or left center to be display symmetry