2.2.2 Descriptive Statistics

Descriptive Statistics Introduction Descriptive statistics summarize and organize data so patterns, central tendencies, and variation become clear. They provide the foundation for understanding a process or system before moving into more advanced statistical methods. Descriptive statistics focus on: - What the data look like - Where the data are centered - How spread out the data are - How the data are shaped - How data sets compare to each other This article covers the concepts and calculations needed to confidently use descriptive statistics in data-driven problem solving. --- Types of Data Qualitative vs Quantitative Data Understanding the type of data is the first step before selecting descriptive measures. - Qualitative (categorical) - Describe categories or labels - Examples: defect type, machine ID, region, color - Summarized by: counts, proportions, mode, frequency tables, bar charts, pie charts - Quantitative (numeric) - Represent measured or counted values - Examples: cycle time, length, weight, cost, number of defects - Summarized by: mean, median, standard deviation, histograms, boxplots, run charts Discrete vs Continuous Data Quantitative data divide further into discrete and continuous. - Discrete data - Counted values, often integers - Examples: number of errors per unit, number of calls, number of defects - Usually come from counting events or items - Continuous data - Measured on a scale with potentially infinite gradations - Examples: time, temperature, length, pressure, weight - Usually come from measurement instruments This classification influences which graphs and statistics are most informative, though many descriptive tools apply to both types. --- Frequency Distributions and Tables Frequency and Relative Frequency A frequency distribution shows how often each value or category occurs. - Frequency: the count of occurrences in each category or class - Relative frequency: frequency divided by total number of observations - Percent frequency: relative frequency × 100 For categorical data: - Create a frequency table listing: - Category - Frequency - Relative frequency (or percent) For numeric data: - Group values into class intervals (bins) - Count how many observations fall into each interval Choosing Class Intervals For continuous data, class intervals should: - Cover the entire range of data - Be mutually exclusive (no overlap) - Be of equal width (for standard histograms) Practical guidelines: - Use between 5–15 classes for most data sets - Class width ≈ (max − min) / number of classes - Round class limits to convenient values --- Graphical Descriptions of Data Histograms A histogram displays the frequency distribution of quantitative data. Key points: - Horizontal axis: class intervals (ranges of values) - Vertical axis: frequency or relative frequency - Bars touch, indicating continuous scale Histograms help identify: - Shape (symmetry or skewness) - Central region - Spread - Potential outliers or gaps - Multiple peaks (modes) Bar Charts and Pie Charts For categorical data: - Bar chart - Horizontal axis: categories - Vertical axis: frequency or percent - Bars separated by space (categories are discrete) - Useful for comparing categories - Pie chart - Circle divided into slices proportional to percent frequency - Emphasizes contribution of each category to the whole - Less precise for detailed comparison than bar charts Stem-and-Leaf Plots Stem-and-leaf plots show the distribution of small to moderate data sets while preserving the actual data values. - Split each observation into: - Stem: leading digit(s) - Leaf: last digit - Example: for 43, 47, 52, 58 - Stems: 4, 5 - Leaves on 4-stem: 3, 7 - Leaves on 5-stem: 2, 8 Benefits: - Show shape of distribution - Maintain individual data values - Quickly identify center and spread Boxplots Boxplots (box-and-whisker plots) summarize distribution using five key numbers. - Elements: - Minimum (not including outliers) - First quartile (Q1) - Median (Q2) - Third quartile (Q3) - Maximum (not including outliers) - Box: from Q1 to Q3 (interquartile range, IQR) - Line inside box: median - Whiskers: extend from box to min and max within a limit (commonly 1.5 × IQR from the quartiles) - Points beyond whiskers: potential outliers Boxplots are powerful for: - Comparing distributions across groups - Visualizing center, spread, skewness, and outliers Run Charts and Time Plots When data are collected over time, plotting in time order reveals patterns. - Run chart - Horizontal axis: time (or sequence) - Vertical axis: measured value - Often includes a reference line (e.g., mean or median) Run charts reveal: - Trends (systematic increases or decreases) - Shifts (sudden changes in level) - Cycles or seasonal patterns - Short-term fluctuations Although often used for process monitoring, run charts are also descriptive tools to understand data behavior across time. --- Measures of Central Tendency Mean The mean is the arithmetic average and is the most widely used measure of center. For a sample of size n: - Sample mean: [ \bar{x} = \frac{\sum{i=1}^{n} xi}{n} ] For a population: - Population mean: [ \mu = \frac{\sum{i=1}^{N} xi}{N} ] Key characteristics: - Uses all data values - Sensitive to extreme values (outliers) - Frequently used in further calculations (e.g., variance, standard deviation) Median The median is the middle value when data are ordered. - If n is odd: median is the middle observation - If n is even: median is the average of the two middle observations Properties: - Resistant to extreme values - Useful when data are skewed or contain outliers - Splits ordered data into two halves of equal count Mode The mode is the most frequently occurring value or category. - For numeric data: - May be unimodal (one clear peak) - May be bimodal or multimodal (two or more peaks) - For categorical data: - Category with the highest frequency The mode is useful for: - Identifying typical categories - Detecting multiple clusters in numeric data Comparing Mean, Median, and Mode Relationship to distribution shape: - Symmetric distribution: - Mean ≈ median ≈ mode - Right-skewed distribution: - Mean > median > mode - Left-skewed distribution: - Mean < median < mode Choice of measure: - Use mean when data are symmetric without extreme outliers - Use median when data are skewed or contain outliers - Use mode for categorical data or when identifying peaks in distribution --- Measures of Variation Range The range is the simplest measure of spread. - Range = maximum − minimum Characteristics: - Easy to compute - Uses only two data points - Highly sensitive to outliers - Gives a rough sense of total spread, but not internal variation Variance Variance measures average squared deviation from the mean. For a sample of size n: - Sample variance: [ s^2 = \frac{\sum{i=1}^{n}(xi - \bar{x})^2}{n - 1} ] For a population: - Population variance: [ \sigma^2 = \frac{\sum{i=1}^{N}(xi - \mu)^2}{N} ] Key features: - Always nonnegative - In squared units of the original data - Forms the basis for standard deviation and many inferential methods Standard Deviation Standard deviation is the square root of variance and is the most commonly used measure of spread. For a sample: - Sample standard deviation: [ s = \sqrt{s^2} ] For a population: - Population standard deviation: [ \sigma = \sqrt{\sigma^2} ] Interpretation: - Measures typical distance of data points from the mean - Expressed in the same units as the original data - Larger standard deviation means greater variability Standard deviation is central for: - Describing dispersion - Comparing variability between groups - Relating data to the normal distribution - Calculating many process-related metrics Coefficient of Variation The coefficient of variation (CV) expresses variation relative to the mean. For sample data: - [ CV = \frac{s}{\bar{x}} \times 100% ] Uses: - Comparing variability across different units or scales - Example: compare variability of time in minutes vs cost in dollars - Comparing variability between groups with different means Interpretation: - Higher CV indicates more variability relative to the mean - Only meaningful when data are on a ratio scale and mean > 0 --- Percentiles and Quartiles Percentiles A percentile indicates the value below which a given percentage of observations fall. - p-th percentile (Pₚ): - Value where p% of data are at or below that value - Examples: - 25th percentile: 25% of values are below this point - 90th percentile: 90% of values are below this point Percentiles are useful for: - Setting performance benchmarks - Describing relative standing (e.g., “in the 95th percentile”) Quartiles Quartiles divide ordered data into four equal parts. - First quartile (Q1): 25th percentile - Second quartile (Q2): 50th percentile (median) - Third quartile (Q3): 75th percentile Quartiles are key to: - Constructing boxplots - Measuring middle spread (via interquartile range) Interquartile Range (IQR) The interquartile range measures spread of the middle 50% of data. - IQR = Q3 − Q1 Characteristics: - Resistant to extreme values - Useful for: - Summarizing middle variability - Identifying potential outliers Common outlier rule using IQR: - Lower fence = Q1 − 1.5 × IQR - Upper fence = Q3 + 1.5 × IQR - Values outside these fences are often flagged as outliers --- Shape of Distributions Symmetry and Skewness Understanding shape helps interpret which measures of center and spread are most meaningful. - Symmetric distribution - Left and right sides mirror each other - Mean and median are close - Example: ideal normal distribution - Right-skewed (positively skewed) - Tail extends to the right (toward larger values) - Mean > median - Often arises from: - Time to complete a task - Cost or income distributions - Data bounded below by zero - Left-skewed (negatively skewed) - Tail extends to the left (toward smaller values) - Mean < median - Can arise from upper-bounded data (e.g., near 100% yields) Modality Modality indicates how many peaks a distribution has. - Unimodal: one main peak - Bimodal: two main peaks - Multimodal: more than two peaks Multiple peaks often indicate: - Mixtures of different subgroups - Changes in process conditions - Need to separate data by relevant factors for clearer analysis Kurtosis (Conceptual) Kurtosis describes how concentrated data are around the mean and in the tails. - High kurtosis (leptokurtic): - More data near mean and heavy tails - More extreme values than a normal distribution - Low kurtosis (platykurtic): - Less data near mean and lighter tails - Fewer extreme values than normal In descriptive work, kurtosis is mostly used conceptually to: - Compare tail behavior to the normal distribution - Indicate potential issues with extreme values --- Descriptive Statistics for Categorical Data Frequency Tables and Proportions For qualitative data, description focuses on counts and proportions. Key measures: - Frequency of each category - Relative frequency (proportion) of each category - Percent frequency of each category These measures reveal: - Most common categories - Rare categories - Overall pattern of occurrence Cross-Tabulations (Contingency Tables) When analyzing two categorical variables together, cross-tabulations show joint frequencies. - Structure: - Rows: categories of one variable - Columns: categories of the other variable - Cells: counts (or percentages) for each combination Uses: - Describing relationships between categorical variables - Comparing category distributions across groups Common summaries: - Row percentages - Column percentages - Overall cell percentages --- Descriptive Statistics for Paired and Multiple Variables Covariance and Correlation (Descriptive View) When examining the relationship between two quantitative variables, descriptive statistics can summarize how they move together. - Covariance (conceptual description): - Positive: variables tend to increase together - Negative: when one increases, the other tends to decrease - Magnitude depends on units, so it is not standardized - Correlation coefficient (r): - Standardized measure of linear association, ranging from −1 to +1 - Values: - Near +1: strong positive linear relationship - Near 0: little or no linear relationship - Near −1: strong negative linear relationship - Unitless, so comparisons across different variables are possible At the descriptive level, correlation is used to: - Summarize strength and direction of linear relationships - Complement scatterplots Scatterplots Scatterplots are primary descriptive tools for two quantitative variables. - Horizontal axis: predictor or input variable - Vertical axis: response or output variable - Each point: one observation’s pair of values Scatterplots help identify: - Direction of relationship (positive, negative, none) - Strength and form (linear, nonlinear) - Outliers or unusual patterns - Clusters or subgroups --- Outliers and Data Integrity Identifying Outliers Outliers are observations that are distant from the rest of the data. Common descriptive indicators: - Points far from others in histograms or scatterplots - Points beyond boxplot whiskers (using IQR fences) - Values more than a certain number of standard deviations from the mean (e.g., beyond ±3s in approximately normal data) Outliers may arise from: - Measurement or recording errors - Changes in process conditions - Rare but genuine extreme events Handling Outliers Before deciding what to do with outliers: - Check for data errors and correct them if found - Investigate context (time, conditions, source) Possible actions: - Keep them if they represent real, meaningful variation - Analyze with and without outliers to assess impact - Document any decision to remove or adjust them Outliers strongly influence: - Mean and standard deviation - Range - Correlation They affect both descriptive summaries and later inferential analyses, so careful consideration is necessary. --- Using Descriptive Statistics Effectively Stepwise Descriptive Approach A systematic descriptive approach typically follows this order: - Understand the data type: - Qualitative vs quantitative - Discrete vs continuous - Visualize the data: - Histograms, boxplots, stem-and-leaf for numeric data - Bar charts, pie charts for categorical data - Run charts or time plots if data are time-ordered - Scatterplots for pairs of quantitative variables - Summarize numerically: - Center: mean, median, mode - Spread: range, variance, standard deviation, IQR - Position: percentiles, quartiles - Relationships: correlation (for paired quantitative variables) - Interpret in context: - Compare groups or time periods - Relate shape, center, and spread to process behavior - Watch for skewness, multimodality, and outliers Comparing Groups Descriptive statistics are essential for comparing different groups or conditions. For each group: - Visualize: - Overlaid histograms - Side-by-side boxplots - Grouped bar charts (for categorical outcomes) - Summarize: - Means and medians - Standard deviations and IQRs - Sample sizes (for understanding stability of estimates) Look for: - Differences in location (center) - Differences in variability - Differences in distribution shape - Presence or absence of outliers in each group --- Summary Descriptive statistics organize and summarize data to reveal key characteristics before deeper analysis. Core elements include: - Understanding data types (qualitative vs quantitative; discrete vs continuous) - Displaying data through histograms, bar charts, boxplots, run charts, stem-and-leaf plots, and scatterplots - Measuring center with mean, median, and mode - Measuring spread with range, variance, standard deviation, interquartile range, and coefficient of variation - Describing position using percentiles and quartiles - Interpreting distribution shape, including symmetry, skewness, modality, and kurtosis - Summarizing relationships between variables with cross-tabulations, covariance, correlation, and scatterplots - Identifying and assessing outliers and their impact Mastering these descriptive tools allows for clear understanding and communication of how data behave, forming a solid foundation for any subsequent statistical or process analysis.