Basic StatisticsBasic Statistics

Six Sigma FoundationsContinuous Improvement TrainingSix Sigma FoundationsContinuous Improvement Training

Six Sigma Simplicity

Key Learning PointsKey Learning Points

s Simple Statistics can:s Increase your Understanding of Process

Behaviors Helps Identify Improvement

Opportunities for 6S

s Simple Statistics can:s Increase your Understanding of Process

Behaviors Helps Identify Improvement

Opportunities for 6S

StatisticsStatisticss Common statistics:

s Miles per gallon (liter); mpg (mpl)s Median home pricess Consumer price indexs Inflation rates Stock market averages Airline on-time arrival rate

s Statistics are computed using data.s Statistics summarize the data and help us

to predict future performance.

s Common statistics:s Miles per gallon (liter); mpg (mpl)s Median home pricess Consumer price indexs Inflation rates Stock market averages Airline on-time arrival rate

s Statistics are computed using data.s Statistics summarize the data and help us

to predict future performance.

Basic StatisticsBasic Statistics

s Serve as a means to analyze data collected in the Measure phase.

s Allow us to numerically describe the data that characterizes our process’ Xs and Ys.

s Use past process and performance data to make inferences about the future.

s Serve as a foundation for advanced statistical problem-solving methodologies.

s Are a concept that creates a universal language based on numerical facts rather than intuition.

s Serve as a means to analyze data collected in the Measure phase.

s Allow us to numerically describe the data that characterizes our process’ Xs and Ys.

s Use past process and performance data to make inferences about the future.

s Serve as a foundation for advanced statistical problem-solving methodologies.

s Are a concept that creates a universal language based on numerical facts rather than intuition.

Data VisualizationData Visualization

s Before any statistical tools are applied, visually display and look at your data.

s A histogram allows us to look at how the data is distributed across our Y scale of measure.

s Before any statistical tools are applied, visually display and look at your data.

s A histogram allows us to look at how the data is distributed across our Y scale of measure.

Number of Wins for National Football League Teams (1998)

151050

5

4

3

2

1

0

Num

ber

of T

eam

s

Five teams won eight games

Source: AOLSports

Number of Games Won

Building a HistogramBuilding a Histogram

The following data came from our bicycle test facility: stopping distances required to bring a 150 lb weight to a complete stop with the rear brake applied from a 10 mph cruising speed.

The following data came from our bicycle test facility: stopping distances required to bring a 150 lb weight to a complete stop with the rear brake applied from a 10 mph cruising speed.

Trial (sample #) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Stop Distance (Feet) 14 6 13 7 10 10 11 9 11 9 11 9 10 10 10

Feet

X-Axis

Y-Axis

Fre

qu

ency

7 8 9 10 11 12 13 146

In addition to counting occurrences and graphing the results, we can describe processes in terms of central tendency and dispersion.

Measures of Central TendencyMeasures of Central Tendency

Measures of Central Tendencys Mean (m, Xbar)—The arithmetic average of a set

of valuess Uses the quantitative value of each data points Is strongly influenced by extreme values

s Median (M)—The number that reflects the middle of a set of valuess Is the 50th percentiles Is identified as the middle number after all the values are

sorted from high to lows Is not affected by extreme values

s Mode—The most frequently occurring value in a data set

Measures of Central Tendencys Mean (m, Xbar)—The arithmetic average of a set

of valuess Uses the quantitative value of each data points Is strongly influenced by extreme values

s Median (M)—The number that reflects the middle of a set of valuess Is the 50th percentiles Is identified as the middle number after all the values are

sorted from high to lows Is not affected by extreme values

s Mode—The most frequently occurring value in a data set

Central Tendency ExerciseCentral Tendency Exercise

s Determine the mean, median, and mode for the bicycle stopping distances used to create the histograms. Mean = ________Median = ________Mode = ________

s Determine the mean, median, and mode for the bicycle stopping distances used to create the histograms. Mean = ________Median = ________Mode = ________

Trial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Stop Distance (Feet) 14 6 13 7 10 10 11 9 11 9 11 9 10 10 10

1201008060

120

80

40

0

Positive Skew

Fre

quen

cy

Mean

Median

Mode

806040200

100

50

0

Negative Skew

Fre

quen

cy

Mean

Median

Mode

11090705030

60

40

20

0

Normal

Fre

quen

cy

ModeMedianMean

Mean, Median, ModeMean, Median, Mode

Range (R)—The difference between the highest and lowest

Sample Variance (s2)—The average squared distance of each point from the average (Xbar)

Sample Standard Deviation(s)—The square root of the variance

Range (R)—The difference between the highest and lowest

Sample Variance (s2)—The average squared distance of each point from the average (Xbar)

Sample Standard Deviation(s)—The square root of the variance

minmax xxR

1

... - -

11

222

2

2

12

n

xxxxxx

n

n

ix

ix

s n

1

1

2

= 2

n

n

ix

ix

ss

Measures of DispersionMeasures of Dispersion

Example of Measures of DispersionExample of Measures of DispersionNumber of Wins for National Football League Teams (1998)

Source: AOLSports

151050

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4

3

2

1

0

Fre

quen

cy

Range = 12

Xbar = 8

s2 = 11.72

s = 3.42

Dispersion ExerciseDispersion ExerciseFind measures of dispersion for the stopping distance data.Fill in the table at the right.

Range (R) = Variance (s2) = Std Dev (s) =

Find measures of dispersion for the stopping distance data.Fill in the table at the right.

Range (R) = Variance (s2) = Std Dev (s) =

A sample is just a subset of all possible values.

PopulationSample

Since the sample does not contain all the possible values, there is some uncertainty about the population. Hence any statistics, such as mean and standard deviation, are just estimates of the true population parameters.

Population vs. Sample(Certainty vs. Uncertainty)Population vs. Sample(Certainty vs. Uncertainty)

Sample Population

Mean(n = # of samples)

StandardDeviation(little “s”)

n

xx

n

ii

1 N

xN

ii

1

1

= 1

2

n

xx

s

n

ii

N

xN

ii

1

2

=

SymbolsSymbols

The Normal CurveThe Normal Curves In 80 to 90% of

problems worked, data will follow a normal bell curve or can be transformed to look like a normal curve.

s This curve is described by the Xbar and s “statistic.”

s The area under this curve is 1 or 100%.

s In 80 to 90% of problems worked, data will follow a normal bell curve or can be transformed to look like a normal curve.

s This curve is described by the Xbar and s “statistic.”

s The area under this curve is 1 or 100%.

s

X

For the normal curve, mean = median = mode.

Normal Bell Curve PropertiesNormal Bell Curve Propertiess Histograms (bar charts) are developed from samples.s Sample statistics (Xbar and s) are calculated from representatives

of the population.s From the histogram and sample statistics, we form a curve that

represents the population from which these samples were drawn.

s Histograms (bar charts) are developed from samples.s Sample statistics (Xbar and s) are calculated from representatives

of the population.s From the histogram and sample statistics, we form a curve that

represents the population from which these samples were drawn.

99.9999998% of the data fallswithin 6 standard deviations

from the mean

6sdX

99.73% of the data falls within 3 standard deviations from

the mean

3sdX

68.26% of the data falls within 1 standard deviation

from the mean

1sdX

0 100 200 300

0

10

20

Fre

quen

cy

1151059585

15

10

5

0

Normal

Fre

quen

cy

80 90 100 110 120

0

5

10 Uniform

Fre

quen

cy

5004003002001000

20

10

0

Exponential

Fre

quen

cy

Other Data DistributionsOther Data Distributions

Log Normal

2019181716151413121110987654321

5

4

3

2

1

0

Fre

quen

cyNormal Curve ExerciseNormal Curve Exercise

s Here is a histogram of the bike stopping distance data. (Xbar = 10 , s = 2)

s Does the histogram appear normal?s Draw vertical lines at 1sd, 2sd, 4sds Discuss

s Here is a histogram of the bike stopping distance data. (Xbar = 10 , s = 2)

s Does the histogram appear normal?s Draw vertical lines at 1sd, 2sd, 4sds Discuss

Basic StatisticsBasic Statistics

Six Sigma FoundationsContinuous Improvement TrainingSix Sigma FoundationsContinuous Improvement Training