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Stat 281: Ch. 2--Presenting Data An engineer, consultant and statistician were driving down a steep mountain road. Suddenly, the brakes failed and the car careened down the road out of control. But half way down, the driver somehow managed to stop the car by running it against the embankment, narrowly avoiding going over a very steep cliff. They all got out, shaken, but otherwise unharmed. The consultant said: "To fix this problem we need to organize a committee, have meetings, write several interim reports and develop a solution through a continuous improvement process." The engineer said: "No! That would take too long, and besides that method has never really worked. I have my trusty penknife here and will take apart the brake system, isolate the problem and correct it." The statistician said: "No - you're both wrong! Let's all push the car back up the hill and see if it happens again. We only have a sample size of 1 here!!"

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Fizzy Cola Sales (Showing first 8 of 50) EmployeeGallons Sold P.P.95.00 S.M.100.75 P.T.126.00 P.U.114.00 M.S.134.25 F.K.116.75 L.Z.97.50 F.E.102.25

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The Goal Display data in ways that elucidate the information contained in them Raw Data actually contains all the information available, but it may not be easy to understand It’s not so much the information available that counts—it’s the information you get out!

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Ranked Fizzy Cola Sales RankEmpl.Gal. SoldRankEmpl.Gal. Sold 1T.T.82.5043R.O.133.25 2A.D.88.5044M.S.134.25 3E.I.91.0045O.U.135.00 4A.S.93.2546G.H.135.50 5.5P.P.95.0047R.T.136.00 5.5E.Y.95.0048A.T.137.00 7L.Z.97.5049O.O.144.00 8T.N.99.5050R.N.148.00

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Viewing Data Directly Ranked Data (aka an Array) –Still contains all the information –Can quickly see range (max and min) –May also easily determine median, quartiles, etc. Stem and Leaf –Arranges ranked data into chart-like form

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Fizzy Cola Stem & Leaf 828 9135579 100234556789 1102344555667889 12124455688 132345567 1448

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More Complex Stem & Leaf (MiniTab Style) Stem-and-Leaf of C1 N=16 Leaf Unit=0.010 1597 1597 460148 460148 (5)6102669 7620247 7620247 36358 36358 1643 1643

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Dot Plot for Fizzy Cola Sales Dot plots display vertically stacked dots for each data value. They tend to bring out any “clustering” behavior in the data. Stem & Leaf and Dot Plots begin to give us a picture of the Distribution of Data.

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Summarized Data Frequency Tables –Grouped or ungrouped –Frequency Distribution –Relative Frequency Distribution Bar Graphs Histogram (Numeric Data Only) Pie Charts Often used for Categorical Data

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Fizzy Cola Frequency Table Number of Employees in each Sales Range Gallons SoldEmployees 80-902 >90-1006 >100-11010 >110-12014 >120-1309 >130-1407 >140-1502

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Histogram of Fizzy Cola Sales

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Constructing a Histogram 1.Identify the high (H) and low (L) scores. Find the range. Range = H - L. 2.Select a number of classes and a class width so that the product is a bit larger than the range. 3.Pick a starting point a little smaller than L. Count from L by the width to obtain the class boundaries. Observations that fall on class boundaries are placed into the class interval to the right. Note: 1.The class width is the difference between the upper- and lower-class boundaries. 2.There is no best choice for class widths, number of classes, or starting points.

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Terms Used With Histograms Symmetrical: The sides of the distribution are mirror images. There is a line of symmetry. Uniform (rectangular): Every value appears with equal frequency. Skewed: One tail is stretched out longer than the other. The direction of skewness is on the side of the longer tail (Positively vs. negatively skewed). J-shaped: There is no tail on the side of the class with the highest frequency. Bimodal: The two largest classes are separated by one or more classes. Often implies two populations are sampled. Normal: The distribution is symmetric about the mean and bell-shaped.

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Bimodal Distribution

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Left-Skewed Distribution Ages of Nuns

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Distribution of Categorical Data Day Number Sold Monday15 Tuesday23 Wednesday35 Thursday11 Friday12 Saturday42 Cars Sold in One Week

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Basic Pie Chart Cars Sold in One Week Pie Charts focus our attention on fractions of the whole, especially for the largest classes.

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Three-D Pie Chart Cars Sold in One Week Three-D Pie Charts are “pretty” but can also be used to distort the image.

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Manipulating 3-D Pie Charts Cars Sold in One Week Changing the angle or turning the pie may affect our perception of size.

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Bar Charts for Categorical Data (Bar charts for categorical data are drawn with bars separated, while bars in histograms touch.) Cars Sold in One Week

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Manipulating Bar Charts Cars Sold in One Week Cutting off the vertical axis distorts our perception of the differences between bars.

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Manipulating Bar Charts Cars Sold in One Week Removal of labels on the vertical axis allows bars to be stretched upward to hide the differences.

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Hmmm… It is proven that the celebration of birthdays is healthy. Statistics show that people who celebrate the most birthdays become the oldest. In earlier times, they had no statistics, so they had to fall back on lies. (Stephen Leacock)

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Measures of Central Tendency Statistics used to locate the middle of a set of numeric data, or where the data is clustered. The term average is often associated with all measures of central tendency (book, not me). The mode for discrete data is the value that occurs with greatest frequency. The modal class of a histogram is the class with the greatest frequency. A bimodal distribution has two high-frequency classes separated by classes with lower frequencies.

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Summation Notation

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The Mean Mean: The “regular” average. The sum of all the values divided by the total number of values. The population mean, , (lowercase Greek mu) is the mean of all x values for the population. It is a parameter of the distribution. We usually cannot measure but would like to estimate its value.

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The Sample Mean The sample mean,, (read x-bar) is the mean of all x values for the sample. It is a statistic. The mean can be greatly influenced by outliers. E.g. Bill Gates moves to town.

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Median Median: The value of the data that occupies the middle position when the data are ranked according to size. The sample median (statistic) may be denoted by “x tilde”:. The population median (parameter), , (uppercase Greek mu), is the data value in the middle of the population. To find the median: 1.Rank the data. 2.Determine the depth of the median. 3.Determine the value of the median.

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Mode Mode: The mode is the value of x that occurs most frequently. Note: If two or more values in a sample are tied for the highest frequency (number of occurrences), there is no mode. Midrange: The number exactly midway between a lowest value data L and a highest value data H. It is found by averaging the low and the high values.

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Dispersion How spread apart are the data? Two populations with the same mean can have very different distributions—would like to take measure spread somehow. Range (max-min) –Values in middle are ignored –Dispersion of middle could be very different Use the idea of deviation from the mean: –MAD –Variance –Standard Deviation

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Deviations from the Mean mean deviations x-values

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Some example data ObsDatax 12 24 35 49 Total

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Calculate the mean ObsDataxMean 125 245 355 495 Total20

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Deviation From the Mean ObsDataxMeanDeviation x- x- 125-3 245 3550 4954 Total20200

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Mean Absolute Deviation (MAD) ObsDataxMeanDeviation x- x- Absolute Deviation 125-33 2451 35500 49544 Sum of Absolute Deviations 8 MAD (divide sum by n) 2

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Formula

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Use of Squared Deviations ObsDataxMeanDeviation x- x- Squared Deviation 125-39 2451 35500 495416 Sum of Squared Deviations: SS(x) 26 Variance (Divide Sum by n-1) 8.67 Standard Deviation (Take Square Root) 2.94

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Sums of Squares The sum of squared deviations is denoted by SS(x) and often called the “Sum of Squares for x.” There are also other notations used, including SS x and S xx

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Variance The Variance is the statistician’s favorite measure of dispersion, but in reports or “everyday use” the standard deviation is more commonly given. The Standard Deviation is the square root of the variance. The Variance may be thought of as the average squared deviation from the mean. For a sample, divide by n-1. For a population, divide by N.

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Formulas

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Formulas

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Example: Find the variance and standard deviation for the data {5, 7, 1, 3, 8}.

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Interpretation of s Need to get a sense of the meaning of different values of dispersion measures. Are units same as data or squared? Empirical Rule: 68%, 95%, 99.7% Test of Normality Range as estimator of s

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z-Scores Also “standardized scores” or just “standard scores.” Expresses a quantity in terms of its distance from the mean in standard deviation units.

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More z-Scores The z-score measures the number of standard deviations away from the mean. z-scores typically range from -3.00 to +3.00. z-scores may be used to make comparisons of raw scores. You can calculate back from z-score to raw data value by using the inverse:

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Percentiles Values of the variable that divide a set of ranked data into 100 equal subsets. –Each set of data has 99 percentiles. –The kth percentile, P k, is a value such that at most k% of the data are smaller than P k and at most (100k)% are larger.

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Procedure for finding P k 1.Rank the n observations, lowest to highest. 2.Compute A = (nk)/100. 3.If A is an integer: d(P k ) = A.5 (depth) P k is halfway between the value of the datum in the Ath position and the value of the next datum. If A is a fraction: d(P k ) = B, the next largest integer. P k is the value of the data in the Bth position. Some programs like Excel also do interpolation

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Quartiles Like percentiles except dividing the data set into 4 equal subsets. The first quartile, Q 1, is the same as the 25 th percentile, and The third quartile, Q 3, is the same as the 75 th percentile. The second quartile is the 50 th percentile, which is the median. Sometimes finding Q 1 and Q 3 is described as finding the medians of the bottom half and top half of the data, respectively.

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Five Number Summary The Min, Q 1, Median, Q 3, and Max Indicate how the data is spread out in each quarter. Interquartile Range is the distance between Q 1 and Q 3. The Midquartile is the average of Q 1 and Q 3, another measure of central tendency.

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Box and Whisker Plots

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Hmmm… What did the Box Plot say to the outlier? “Don’t you dare get close to my whisker!”

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