Tool 66: Descriptive Statistics

AKA

N/A

Classification

Planning/Presenting (PP)

Tool description

Descriptive statistics are the procedure used to organize, summarize, and describe data that were collected over time or as a result of a data collection process during a research project. The statistical calculations provide measures of central tendency and measures of dispersion, which are often used as supporting information in the decision-making process.

Typical application

• To summarize research data into meaningful forms using standard statistical techniques.

• To describe, in statistical terms, collected data for reporting purposes.

• To use statistical data to improve the decision-making process.

Problem-solving phase

	Select and define problem or opportunity
	Identify and analyze causes or potential change
	Develop and plan possible solutions or change
→	Implement and evaluate solution or change
→	Measure and report solution or change results
	Recognize and reward team efforts

Typically used by

1	Research/statistics
	Creativity/innovation
2	Engineering
3	Project management
4	Manufacturing
	Marketing/sales
	Administration/documentation
	Servicing/support
	Customer/quality metrics
	Change management

links to other tools

before

• Data Collection Strategy

• Sampling Method

• Random Numbers Generator

• Standard Deviation

• Frequency Distribution (FD)

after

• Process Capability Ratios

• Control Chart

• Correlation Analysis

• Analysis of Variance

• Hypothesis Testing (Chi-Square)

Notes and key points

The normal probability distribution is a symmetrical, bell-shaped distribution frequently used in statistical analyses. The arithmetic mean () median () and mode () are of equal value and are located at the center and peak of the curve. These measures are averages and therefore considered measures of central tendency. Measures of dispersion are measures under the curve, moving horizontally left or right to identify area or probability. Among others, standard deviations (S), z-values (z), and percentiles (%) are most often used in descriptive and inferential statistics.

Symbols and Definitions:

Population	Sample
N = Population size	n = Sample size
μ = Population mean	= Sample mean
σ2 = Population variance	S2 = Sample variance
σ = Population standard deviation	S = Sample standard deviation

Variance: σ² (sigma squared) is the average of the squared deviation scores (x) of the observations from the mean ().

Standard deviation: σ (Sigma) is the square root of the average of the squared deviation scores (x) of the observations from the mean ().

Deviation scores: (x) Found by subtracting the mean () from each observation or data value.

X, Y = Any observation, measurement, data value, or score (X for the first data set, Y for the second data set.)

x, y = Deviation score found by: x = X −

= The Mean: The arithmetic average.

= The Median: The midpoint observation or value of observations that have been ordered from low to high. Half of the observations are above and half of the observation are below the median, or middle observation.

= The Mode: The values of the observation that appear most frequently in a data set.

R = The Range: Highest value-lowest value of observations in a data set.

Z = The z-score: The distance between a specific value of an observation (X) and the mean () divided by the standard deviation (S). The z value is used to access the "Proportions of Area Under the Normal Curve" table to determine the area or probability. (See Appendix, Table A.)

Σ = The sum of (the summation sign).

SK = Skewness: to check if distribution has positive or negative skewness.

Equations:

Note

n−1 is used when total observations = ≤30.

Step-by-step procedure

• STEP 1 The raw data are organized by sorting, sequencing, grouping, or placing them into matrices or tables.

• STEP 2 Basic measures of central tendency, measures of dispersion, and z-scores to percentiles under the normal probability curve can now be calculated. See example Document Wire Extension Measurement.

• STEP 3 All calculations are checked and used in reports or presentations as supporting information.

Example of tool application

Document Wire Extension Measurements

Refer to #8 (notes and key points) for the equations used below.

• Refer to "Percentage of Area Under the Normal Curve" in Table A of the Appendix to check percentage (%); also described in "Normal Probability Distribution" in this book.