Unit 4: Beginner Statistics

Based on the Council for Six Sigma Certification Body of Knowledge

Graphical Analysis

A critical part of the Six Sigma approach. Whenever data or ideas can be displayed in a visual format, obvious data trends can be quickly identified and communicated. Visual analysis is extremely helpful when presenting ideas to auxiliary staff or executives, who might not be trained in intermediate or advanced statistics.

Bar Charts Benefits

Recognizable and easy to read; can summarize large data sets; clarify trends; most people can easily estimate important values; can visually check data and identify areas where data might be skewed; can easily display data sets that range above and below zero on the same chart.

Bar Charts Data Requirements

Typically require nominal or ordinal data - data that is classified according to qualitative information. When displaying nominal data, it's often beneficial to create a Pareto-style chart.

Column vs. Bar Charts

Technically, what is commonly referred to as a bar chart is a column chart where visual columns rise vertically. A literal bar chart displays the same information horizontally.

Stacked Bar Charts

Let you display the total nominal or ordinal data for each category while also breaking that information into color coded categories.

Pie Chart

A visual tool that almost any employee will be familiar with. Ordinal and nominal information can be displayed in a pie chart with the main purpose of visually representing how each category relates to the whole. Conventional wisdom says to use a pie chart when the numbers add up to 100, but this doesn't have to be the case.

Pie Chart Benefits

Can summarize large amounts of specific data types in a visual format; simplicity; can identify obvious problems; usually require very little extra explanation; display the relevance of subset data within a total data set.

Pie Chart Disadvantages

Can be manipulated; usually fail to easily display changes over time; it can be difficult to visualize exact values.

Pie of Pie (Excel Function)

Lets you create an overall pie chart and carve out a second pie chart to delve deeper into certain areas of the larger structure.

XY Scatter Diagrams

Beneficial because they can help teams visually see the relationship between two factors in a process. Scatter diagrams typically help teams see whether there is no correlation, weak correlation, or positive or negative correlation.

Positive Correlation (Scatter Diagram)

Occurs when variable 2 is related to an increase in variable 1, or vice versa.

Negative Correlation (Scatter Diagram)

Occurs when variable 2 is related to a decrease in variable 1, or vice versa.

Scatter Diagram Drawbacks

Not as familiar to business employees as bar, pie, or line charts; risk that individuals will mistake correlation for causation, which can lead to incorrect decisions.

X-Bar Control Chart

Plots the mean of a sample over time – or the mean of samples taken over time, in the case of an active process. X-bar control charts are one of the most frequently used control charts.

Analysis ToolPak (Excel)

A free data analysis tool available from Microsoft for Excel. You can find it under File > Options > Add-ins > Manage Excel Add-ins > Analysis ToolPak.

Descriptive Statistics

Numbers used to describe a set of data. Averages, means, and even standard deviations associated with that data are all descriptive in nature.

Inferential Statistics

Involve calculations and resulting numbers that draw more general conclusions from data. Inferential statistics are used to draw conclusions about the population from sample data.

Probability

The likelihood that a certain event or outcome will occur given a specific set of data or a specific action. Basic probability can be calculated by the formula

Histograms

Bar charts where each bar, or "bin," corresponds to a data range. Data points within a sample or population are divided between the bins and are graphed accordingly.

Histogram Uses

Used to analyze continuous or variable data that is finite in nature or comes from population sampling. Histograms help Six Sigma experts understand how the spread, or distribution, of the data is shaped and where the center of that data might be.

Bi-Modal Distribution (Histogram)

When two specific concentrations, or humps, are noted on a histogram. Often, when you see this shape of data, it means that you are measuring two processes that you believe are a single process.

Skewed Distribution (Histogram)

Features data that is clustered at only one end of the graph. Skewed histograms are common with certain types of data, particularly when data "falls off" at a natural cut-off time.

Number of Bins (Histogram)

The number of bins used is important. If you have too few bins, the graph doesn't provide any relevant information. If you have too many bins, you lose informative functionality and the graph becomes something that looks like a comb.

Normal Distribution (Gaussian Distribution)

Probably the most important distribution related to continuous data from a statistical analysis standpoint. Normal data is shaped symmetrically surrounding the mean.

Normal Distribution Percentages

68.26 percent of all data points fall within plus or minus one standard deviation from the mean; 95.46 percent fall within plus or minus two standard deviations; 99.73 percent fall within plus and minus three standard deviations.

Normality Testing - Chi-Squared Goodness-of-Fit Test

A hypothesis test where

Chi-Squared Goodness-of-Fit Test - Observed Bins

Created from the histogram data table generated by the Analysis ToolPak.

Chi-Squared Goodness-of-Fit Test - Expected Bins

Use statistics related to the normal curve to calculate how we might expect bins to behave given the median and standard deviation of our sample. Use the CDF (Cumulative Distribution Function) to calculate expected bin frequencies.

Cumulative Distribution Function (CDF)

Measures the total area under a curve to the left of the point we are measuring from.

Excel NORMDIST Formula

=NORMDIST(x value, Sample Mean, Sample Standard Deviation, TRUE) - used to calculate CDF. =NORMDIST(x, mean, standard_dev, cumulative) - used to calculate normal probabilities.

Degrees of Freedom (Chi-Squared Test)

bins – 1 - #calculated parameters.

Excel NORMINV Formula

=NORMINV(probability, mean, standard_dev) - used for inverse calculation of normal probabilities.

Correlation Coefficient (R Value)

The statistical calculation that measures correlation. It is a number between -1 and 1. As the correlation coefficient approaches either -1 or 1, a relationship between two variables is considered more likely or stronger.

Correlation Coefficient Equal to 1

The relationship is certain

Correlation Coefficient Equal to -1

All the data points are found on the negative trend line.

Correlation Coefficient Equal to 0

There is certainly no relationship between the variables, and there is no trend line.

Correlation Threshold

In most applications, correlation is considered to occur at some level if the R value is 0.4 or more or -0.4 or less.

Excel CORREL Formula

=CORREL(Array 1, Array 2) - one of the easiest ways to calculate the correlation coefficient.

Coefficient of Determination (r²)

Literally, the square of R, or the square of the correlation coefficient. The r² value tells us how much the fluctuation in one variable (the dependent variable, or y) is related to changes in another variable (the independent variable, or x).

Linear Regression Analysis

Creates a linear model that helps you predict how one variable might behave given changes in the other variable. Regression analysis lets us create a best fit line and equation for our data, and the r² value tells us how confident we can be in using that line to make predictions.

Regression Data Requirements

Both correlation and regression work with continuous or ratio data, and both sets of variables have to be quantitative in nature. Regression requires data that can be written as a function