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Descriptive Statistics

Definition: Sample Mean

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Population Mean

For a finite population with N measurements,the mean is

The sample mean is a reasonable estimate of thepopulation mean .

Descriptive Statistics

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Population Variance

When the population is finite and consists of Nvalues, we may define the population variance as

Descriptive Statistics

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Definition: Sample Variance

The sample variance is a reasonable estimate of the population variance .

Descriptive Statistics

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Stem-and-Leaf Diagrams

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Figure

Stem: Tensdigits. Leaf:

Ones digits.

Stem-and-Leaf Diagrams

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Data Features The median is a measure of central tendency that divides thedata into two equal parts, half below the median and half above.If the number of observations is even, the median is halfwaybetween the two central values.

Stem-and-Leaf Diagrams

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For ordered set of data,

The first or lower quartile , q 1 , is a value that has approximatelyone-fourth (25%) of the observations below it and approximately75% of the observations above.

The second quartile , q 2 , has approximately one-half (50%) of the observations below its value. The second quartile is exactly equal to the median .

The third or upper quartile , q 3, has approximately three-fourths(75%) of the observations below its value. As in the case of themedian, the quartiles may not be unique.

Stem-and-Leaf Diagrams

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Data Features

In general, the 100 k th percentile is a data value such thatapproximately 100 k % of the observations are at or below thisvalue and approximately 100(1 - k )% of them are above it.

Stem-and-Leaf Diagrams

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Frequency Distributions and Histograms

Figure Histogram of compressive strength for 80 aluminum-lithium alloy

specimens.

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Histograms for symmetric and skewed distributions.

Frequency Distributions and Histograms

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6-4 Box Plots

The box plot is a graphical display thatsimultaneously describes several importantfeatures of a data set, such as center, spread,departure from symmetry, and identification of observations that lie unusually far from the bulkof the data.

Whisker Outlier Extreme outlier

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6-4 Box Plots

Figure 6-13 Description of a box plot.

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A Probability Plot

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A Probability Plot

Suppose now that for percentages 100( i .5)/ n for i =1,, n the percentiles are determined for a specifiedpopulation distribution whose plausibility is beinginvestigated.

If the sample is actually conforms the specified distribution,the sample percentiles should be close to thecorresponding population distribution percentiles.

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A Probability Plot

Similarly,

([100( i .5)/ n]th z percentile, i th smallest observation)

on a two-dimensional coordinate system is called a normalprobability plot. If the sample observations are in factdrawn from a normal distribution with mean value and standard deviation, the points should fall close to a straightline with slope and intercept .

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Example 30

The sample consisting of n = 20 observations on dielectricbreakdown voltage of a piece of epoxy resin appeared inthe article

..

The values of ( i .5)/ n for which z percentiles are neededare (1 .5)/20 = .025, (2 .5)/20 = .075,, and .975.

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Example 30

Figure 4.35 shows the resulting normal probability plot. Thepattern in the plot is quite straight, indicating it is plausiblethat the population distribution of dielectric breakdownvoltage is normal.

cont

d

Figure 4.35

Normal probability plot for the dielectric breakdown voltage sample