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Review of Descriptive Graphs and Measures Here is a quick review of what we have covered so far. Pie Charts Bar Charts Pareto Tables Dotplots Stem-and-leaf Histograms Ogives Boxplots Time Series Mean Median Mode Weighted mean Range IQR Variance and Standard Deviation Mean and Standard Deviation of a frequency distribution Median of a distribution Empirical Rule Z-scores Quartiles, percentiles

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Review of Descriptive Graphs and Measures Here are some ways to display Categorical Data: Pareto Chart Bar Graph Pie Chart Categoryfrequencypercentcumulative tardies1336.11%13 no id925.00%22 talk616.67%28 walk411.11%32 throw38.33%35 lie12.78%36 total361.00%

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Review of Descriptive Graphs and Measures 6 | 7 7 | 1 8 8 | 2 5 6 7 7 9 | 2 5 7 9 9 10 | 0 1 2 3 3 4 5 5 7 8 9 11 | 2 6 8 12 | 2 4 5 Stem-and-leafLow/High Stem-and-leaf 6 | 7 7 | 1 7 | 8 8 | 2 8 | 5 6 7 7 9 | 2 9 | 5 7 9 9 10 | 0 1 2 3 3 4 10 | 5 5 7 8 9 11 | 2 11 | 6 8

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Review of Descriptive Graphs and Measures 66768696106116126 Phone minutes Dot-plot or Line Plot

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Review of Descriptive Graphs and Measures 5545352515 4245 30 17 55 Interquartile Range = 45 – 30 = 15 Min Q1 median Q3 max

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Review of Descriptive Graphs and Measures AbsencesGrade 02468101214 16 40 45 50 55 60 65 70 75 80 85 90 95 Absences (x) x 8 2 5 12 15 9 6 y 78 92 90 58 43 74 81 Final grade (y) Scatterplot

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Measures of Central Tendency The mode is the value that occurs the most. There can be more than one mode. The median is the middle value in an ordered data set The arithmetic mean is the center of gravity of the data set. This is obtained by summing all of the values and dividing by the number of values.

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Measures of Central Tendency We can also find the mean of a frequency distribution by calculating. This is usually easier to do with a table: xfxf 212 3412 4624 5210 616 1454 Mean = 54/14 3.86

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Measures of Central Tendency For classes containing multiple values, you use the midpoint of the class as the x. Classmidpointfxf 0-1.9111 2-3.93412 4-5.95630 6-7.97214 8-9.9919 1466 Mean =66/14 4.71

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Measures of Central Tendency The class with the highest frequency is called the modal class. modal class Classmidpointfxf 0-1.9111 2-3.93412 4-5.95630 6-7.97214 8-9.9919 1466

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Measures of Central Tendency We estimate the median as the midpoint of the class it lies in. Classmidpointfxf 0-1.9111 2-3.93412 4-5.95630 6-7.97214 8-9.9919 1466 median lies in here, so we estimate the median as 5.

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Measures of Central Tendency Finally, there is the weighted mean: xweightxw Tests86.543 Classwork /homewk 90.2522.5 Quizzes76.2519 84.5

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Measures of Variation The range is the largest value minus the smallest value The Interquartile range is the Third Quartile minus the First Quartile

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Measures of Variation The Variance is : Example data set one: 1, 3, 5, 7, 8, 9, 9, 11, 12, 12, 15 The mean is about 8.36 The variance is [(1-8.36 )2+( 3-8.36 )2+( 5-8.36 )2+( 7- 8.36 )2+( 8-8.36 )2+( 9-8.36 )2+( 9-8.36 )2+( 11-8.36 )2+( 12- 8.36 )2+( 12-8.36 )2+( 15-8.36 )2 ] / 11=3.98 The Standard Deviation is the square root of the Variance.

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Measures of Variation The standard deviation is the easier to find using a calculator with the function built in. Example data set one: 1, 3, 5, 7, 8, 9, 9, 11, 12, 12, 15 TI-83: Put the data in L1 Press Stat. Cursor right to choose Calc. Enter for one variable stats. The mean, standard deviation and several other measures will be displayed.

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Measures of Variation The standard deviation can be calculated using a table. xfxfx2fx2f 2124 341236 462496 521050 61636 1454222

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Measures of Variation However the calculator is still probably easier: xf 21 34 46 52 61 14 TI-83 Enter values in L1 Enter frequencies in L2 One-variable-stats L1, L2 This will give you the standard deviation of the frequency table

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Empirical Rule The Empirical Rule for Normal Distributions About 68% of all values fall within 1 standard deviation of the mean About 95% of all values fall within 2 standard deviation of the mean About 99.7% of all values fall within 3 standard deviation of the mean.

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Empirical Rule The Empirical Rule for Normal Distributions About 68% of all values fall within 1 standard deviation of the mean About 95% of all values fall within 2 standard deviation of the mean About 99.7% of all values fall within 3 standard deviation of the mean.

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Empirical Rule The Empirical Rule for Normal Distributions About 68% of all values fall within 1 standard deviation of the mean About 95% of all values fall within 2 standard deviation of the mean About 99.7% of all values fall within 3 standard deviation of the mean. Example: A normal dataset has a mean of 50 and a standard deviation of 5. Between what two numbers does 95% of the data fall? (50-2*5, 50+2*5) (40, 60)

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Percentiles Count the number of data points that lie below the value Divide this by the total number of data points Convert to a percent (multiply by 100) Reading a percentile chart:

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Z-scores Z-score: The number of standard deviations a data point is from the mean. Find the raw distance from the mean. Divide by the standard deviation. Example: A data set has a mean of 50 and a SD of 5. What is the z-score of 62? Z = (x – mean)/SD = (62 – 50)/5 = 12/5 = 2.4

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Measures of Variation Z-score The number of standard deviations a data point is from the mean.

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