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CHAPTER OUTLINE

6-1 Numerical Summaries of Data 6-2 Stem-and-Leaf Diagrams 6-3 Frequency Distributions and Histograms

6-4 Box Plots 6-5 Time Sequence Plots 6-6 Probability Plots

Chapter 6 Title and Outline 1

Numerical Summaries of Data • Data are the numeric observations of a phenomenon of interest. The totality of all observations is a population. A portion used for analysis is a random sample. • We gain an understanding of this collection, possibly massive, by describing it numerically and graphically, usually with the sample data. • We describe the collection in terms of shape, center, and spread. • The center is measured by the mean. • The spread is measured by the variance. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

2

Populations & Samples

Figure 6-3 (out of order) A population is described, in part, by its parameters, i.e., mean (μ) and standard deviation (σ). A random sample of size n is drawn from a population and is described, in part, by its statistics, i.e., mean (x-bar) and standard deviation (s). The statistics are used to estimate the parameters. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

3

Mean If the n observations in a random sample are denoted by x1 , x2 ,..., xn , the sample mean is n

∑x

x1 + x2 + ... + xn i =1 i x= = (6-1) n n For the N observations in a population population denoted by x1 , x2 ,..., xN , the population mean is analogous to a probability distribution as N

N

µ = ∑ xi ⋅ f ( x ) = i =1

∑x

i

i =1

N

Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

(6-2) 4

Example 6-1: Sample Mean Consider 8 observations (xi) of pull-off force from engine connectors as shown in the table. 8

∑x

i

x = average =

i =1

8

12.6 + 12.9 + ... + 13.1 = 8

104 = = 13.0 pounds 8

i 1 2 3 4 5 6 7 8

xi 12.6 12.9 13.4 12.2 13.6 13.5 12.6 13.1 12.99 = AVERAGE($B2:$B9)

Figure 6-1 The sample mean is the balance point. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

5

Variance Defined If the n observations in a sample are denoted by x1 , x2 ,..., xn , the sample variance is n

∑ ( x − x)

2

i

s2 =

i =1

(6-3)

n −1

For the N observations in a population denoted by x1 , x2 ,..., xN , the population variance, analogous to the variance of a probability distribution, is N

N

σ 2 = ∑ ( xi − µ ) ⋅ f ( x ) = i =1

2

∑ ( xi − µ )

2

i =1

N

Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

(6-5) 6

Standard Deviation Defined • The standard deviation is the square root of the variance. • σ is the population standard deviation symbol. • s is the sample standard deviation symbol. • The units of the standard deviation are the same as: – The data. – The mean. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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Rationale for the Variance

Figure 6-2 The xi values above are the deviations from the mean. Since the mean is the balance point, the sum of the left deviations (negative) equals the sum of the right deviations (positive). If the deviations are squared, they become a measure of the data spread. The variance is the average data spread. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

8

Example 6-2: Sample Variance Table 6-1 displays the quantities needed to calculate the summed squared deviations, the numerator of i xi x i - xbar (x i - xbar) 2 the variance. Dimension of: xi is pounds Mean is pounds. Variance is pounds2. Standard deviation is pounds. Desired accuracy is generally accepted to be one more place than the data.

1 2 3 4 5 6 7 8 sums =

12.6 -0.40 0.1600 12.9 -0.10 0.0100 13.4 0.40 0.1600 12.3 -0.70 0.4900 13.6 0.60 0.3600 13.5 0.50 0.2500 12.6 -0.40 0.1600 13.1 0.10 0.0100 104.00 0.00 1.6000 divide by 8 divide by 7 mean = 13.00 variance = 0.2286 standard deviation = 0.48

Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

9

Computation of s2 The prior calculation is definitional and tedious. A shortcut is derived here and involves just 2 sums. n

∑ ( x − x)

2

n

i

s2 =

i =1

n

∑x

2 i

=

=

n −1

∑( x

2 i

2

+ x − 2 xi x

i =1

n −1 n

n

+ nx − 2 x ∑ xi 2

i =1

i =1

n −1

∑x

2 i

=

2 i

2

n

+ nx − 2 x ⋅ nx n −1

n

2 i

2

i =1

x − ∑ xi x − nx ∑ ∑ i =1 = i =1 = i =1 n −1 n −1 n

)

2

n

Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

(6-4) 10

Example 6-3: Variance by Shortcut x − ∑ xi ∑ i =1 i =1 2 s = n −1 n

n

2 i

2

n

2

1,353.60 − (104.0 ) 8 = 7 1.60 = = 0.2286 pounds 2 7

i 1 2 3 4 5 6 7 8 sums =

xi 12.6 12.9 13.4 12.3 13.6 13.5 12.6 13.1 104.0

x i2 158.76 166.41 179.56 151.29 184.96 182.25 158.76 171.61 1,353.60

s = 0.2286 = 0.48 pounds Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

11

What is this “n–1”? • The population variance is calculated with N, the population size. Why isn’t the sample variance calculated with n, the sample size? • The true variance is based on data deviations from the true mean, μ. • The sample calculation is based on the data deviations from x-bar, not μ. X-bar is an estimator of μ; close but not the same. So the n-1 divisor is used to compensate for the error in the mean estimation. Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

12

Degrees of Freedom • The sample variance is calculated with the quantity n-1. • This quantity is called the “degrees of freedom”. • Origin of the term: – There are n deviations from x-bar in the sample. – The sum of the deviations is zero. (Balance point) – n-1 of the observations can be freely determined, but the nth observation is fixed to maintain the zero sum.

Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

13

Sample Range If the n observations in a sample are denoted by x1, x2, …, xn, the sample range is: r = max(xi) – min(xi) It is the largest observation in the sample less the smallest observation. From Example 6-3: r = 13.6 – 12.3 = 1.30 Note that: population range ≥ sample range Sec 6-1 Numerical Summaries of Data © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

14

Intro to Stem & Leaf Diagrams First, let’s discuss dot diagrams – dots representing data on the number line. Minitab produces this graphic using the Example 6-1 data. Dotplot of Force

12.4

12.6

12.8

13.0 Force

13.2

13.4

13.6

Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

15

Stem-and-Leaf Diagrams • Dot diagrams (dotplots) are useful for small data sets. Stem & leaf diagrams are better for large sets. • Steps to construct a stem-and-leaf diagram: 1) Divide each number (xi) into two parts: a stem, consisting of the leading digits, and a leaf, consisting of the remaining digit. 2) List the stem values in a vertical column (no skips). 3) Record the leaf for each observation beside its stem. 4) Write the units for the stems and leaves on the display. Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

16

Example 6-4: Alloy Strength Table 6-2 Compressive Strength (psi) of Aluminum-Lithium Specimens 105 221 183 186 121 181 180 143 97 154 153 174 120 168 167 141 245 228 174 199 181 158 176 110 163 131 154 115 160 208 158 133 207 180 190 193 194 133 156 123 134 178 76 167 184 135 229 146 218 157 101 171 165 172 158 169 199 151 142 163 145 171 148 158 160 175 149 87 160 237 150 135 196 201 200 176 150 170 118 149

Figure 6-4 Stem-and-leaf diagram for Table 6-2 data. Center is about 155 and most data is between 110 and 200. Leaves are unordered. Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

17

Stem-and-Leaf Diagrams • Table 6-2 data: Leaves are ordered, hence the data is sorted. • Median is the middle of the sorted observations. – If n is odd, the middle value. – If n is even, the average or midpoint of the two middle values. Median is 161.5.

• Mode is 158, the most frequent value. Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

18

Quartiles • The three quartiles partition the data into four equally sized counts or segments. – 25% of the data is less than q1. – 50% of the data is less than q2, the median. – 75% of the data is less than q3.

• Calculated as Index = f (n+1) where: – Index (I) is the Ith item (interpolated) of the sorted data list. – f is the fraction associated with the quartile. – n is the sample size. Value of • For the Table 6-2 data: indexed item f Index I th (I +1)th quartile 0.25 20.25 143 145 143.50 0.50 40.50 160 163 161.50 0.75 60.75 181 181 181.00 Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

19

Percentiles • Percentiles are a special case of the quartiles. • Percentiles partition the data into 100 segments. • The Index = f (n+1) methodology is the same. • The 37%ile is calculated as follows: – Refer to the Table 6-2 stem-and-leaf diagram. – Index = 0.37(81) = 29.97 – 37%ile = 153 + 0.97(154 – 153) = 153.97

Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

20

Interquartile Range • The interquartile range (IQR) is defined as: IQR = q3 – q1. • From Table 6-2: IQR = 181.00 – 143.5 = 37.5 Variable Strength

N 80 Min 76.00

Mean 162.66

StDev 33.77

Q1 Median Q3 143.50 161.50 181.00 5-number summary

Max 245.00

Sec 6-2 Stem-And-Leaf Diagrams © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

21

Histograms • A histogram is a visual display of the data distribution, similar to a bar chart or a stemand-leaf diagram. • Steps to build one with equal bin widths: 1) Label the bin boundaries on the horizontal scale. 2) Mark & label the vertical scale with the frequencies or relative frequencies. 3) Above each bin, draw a rectangle whose height is equal to the frequency or relative frequency. Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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Histograms Frequency Distribution for the data in Table 6-2

Considerations: Range = 245 – 76 = 169 Sqrt(80) = 8.9 Trial class width = 18.9

Decisions: Number of classes = 9 Class width = 20

Table 6-4 Frequency Distribution of Table 6-2 Data Cumulative Relative Relative Class Frequency Frequency Frequency 70 ≤ x < 90 2 0.0250 0.0250 90 ≤ x < 110 3 0.0375 0.0625 110 ≤ x < 130 6 0.0750 0.1375 130 ≤ x < 150 14 0.1750 0.3125 150 ≤ x < 170 22 0.2750 0.5875 170 ≤ x < 190 17 0.2125 0.8000 190 ≤ x < 210 10 0.1250 0.9250 210 ≤ x < 230 4 0.0500 0.9750 230 ≤ x < 250 2 0.0250 1.0000 80 1.0000

Range of classes = 20 * 9 = 180 Starting point = 70 Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

23

Histogram of the Table 6-2 Data

Figure 6-7 Histogram of compressive strength of 80 aluminum-lithium alloy specimens. Note these features – (1) horizontal scale bin boundaries & labels with units, (2) vertical scale measurements and labels, (3) histogram title at top or in legend. Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

24

Cumulative Frequency Plot

Figure 6-10 Cumulative histogram of compressive strength of 80 aluminum-lithium alloy specimens. Comment: Easy to see cumulative probabilities, hard to see distribution shape. Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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Shape of a Frequency Distribution

Figure 6-11 Histograms of symmetric and skewed distributions. (b) Symmetric distribution has identical mean, median and mode measures. (a & c) Skewed distributions are positive or negative, depending on the direction of the long tail. Their measures occur in alphabetical order as the distribution is approached from the long tail.☺

Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

26

Histograms for Categorical Data • Categorical data is of two types: – Ordinal: categories have a natural order, e.g., year in college, military rank. – Nominal: Categories are simply different, e.g., gender, colors.

• Histogram bars are for each category, are of equal width, and have a height equal to the category’s frequency or relative frequency. • A Pareto chart is a histogram in which the categories are sequenced in decreasing order. This approach emphasizes the most and least important categories. Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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Example 6-6: Categorical Data Histogram

Figure 6-12 Airplane production in 1985. (Source: Boeing Company) Comment: Illustrates nominal data in spite of the numerical names, categories are shown at the bin’s midpoint, a Pareto chart since the categories are in decreasing order. Sec 6-3 Frequency Distributions And Histograms © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

28

Box Plot or Box-and-Whisker Chart • A box plot is a graphical display showing center, spread, shape, and outliers. • It displays the 5-number summary: min, q1, median, q3, and max.

Figure 6-13 Description of a box plot. Sec 6-4 Box Plots

29 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Box Plot of Table 6-2 Data

Figure 6-14 Box plot of compressive strength of 80 aluminumlithium alloy specimens. Comment: Box plot may be shown vertically or horizontally, data reveals three outliers and no extreme outliers. Lower outlier limit is: 143.5 – 1.5*(181.0-143.5) = 87.25. Sec 6-4 Box Plots

30 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Time Sequence Plots • A time series plot shows the data value, or statistic, on the vertical axis with time on the horizontal axis. • A time series plot reveals trends, cycles or other time-oriented behavior that could not be otherwise seen in the data.

Figure 6-16 Company sales by year (a) & by quarter (b). The annual time interval masks cyclical quarterly variation, but shows consistent progress. Sec 6-5 Time Sequence Plots © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

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Probability Plots • How do we know if a particular probability distribution is a reasonable model for a data set? • We use a probability plot to verify such an assumption using a subjective visual examination. • A histogram of a large data set reveals the shape of a distribution. The histogram of a small data set would not provide such a clear picture. • A probability plot is helpful for all data set sizes. Sec 6-6 Probability Plots

32 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

How To Build a Probability Plot • To construct a Normal Probability Plot (NPP): – Sort the data observations in an ascending order: x(1), x(2),…, x(n). – The observed value x(j) is plotted against the cumulative distribution (j – 0.5)/n. – If the paired numbers form a straight line, it is reasonable to assume that the data follows the proposed distribution. Sec 6-6 Probability Plots

33 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Example 6-7: Battery Life The effective service life (minutes) of batteries used in a laptop are given in the table. We hypothesize that battery life is adequately modeled by a normal distribution. The probability plot is shown on normal probability vertical scale. Table 6-6 Calculations for Constructing a Normal Probability Plot j 1 2 3 4 5 6 7 8 9 10

x (j ) 176 183 185 190 191 192 201 205 214 220

(j -0.5)/10 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

100(j -0.5)/10 5 15 25 35 45 55 65 75 85 95

Figure 6-22 Normal probability plot for battery life.

Sec 6-6 Probability Plots

34 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Use of the Probability Plot • The probability plot can identify variations from a normal distribution shape. – Light tails of the distribution – more peaked. – Heavy tails of the distribution – less peaked. – Skewed distributions.

• Larger samples increase the clarity of the conclusions reached.

Sec 6-6 Probability Plots

35 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Probability Plot Variations

Figure 6-24 Normal probability plots indicating a non-normal distribution. (a) Light tailed distribution (b) Heavy tailed distribution (c) Right skewed distribution

Sec 6-6 Probability Plots

36 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.

Important Terms & Concepts of Chapter 6 Standard deviation Box plot Variance Frequency distribution & Probability plot histogram Relative frequency Median, quartiles & distribution percentiles Sample: Multivariable data Mean Normal probability plot Standard deviation Pareto chart Variance Population: Stem-and-leaf diagram Mean Time series plots Chapter 6 Summary

37 © John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.