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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Do Now: Aim: How do we use measures of dispersion: range, variance, and standard deviation? Find the average of each set 1 2 3 4 5 average does not give sufficient info about data

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Recall: Model Problem A teacher marked a set of 32 papers. The grades were as follows: 90, 85, 74, 86, 65, 62, 100, 95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73, 100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72, 75. STAT EDIT 1 STAT CALC 1 ENTER

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Range Range – the difference between the highest value and the lowest value in a set of data. A teacher marked a set of 32 papers. The grades were as follows: 90, 85, 74, 86, 65, 62, 100, 95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73, 100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72, 75. Range – 100 – 45 = 55 Often unreliable as a measure of dispersion

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Quartiles Find the lower and upper quartile of Range - lowest to highest amount Q3Q1Median 25%50%75% Quartiles break a data group into 4 equal parts. The lower quartile is the median of the lower half. The upper quartile is median of the upper half. Paper Grade Problem n = 32 Lower quartile is the median of the first 16 numbers Upper quartile is the median of the last 16 numbers (86 + 89)/2 = 87.5 Average of the 8th & 9th numbers Average of the 24th & 25th numbers (70 + 72)/2 = 71 7810045 71 87.5

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Percentiles Q3 Q1 Median Lower quartile 25% Second quartile 50% Upper quartile 75% Percentile is a number that tells us what percent of the total number of data values lie at or below a given measure. Ranking Paper Grade Problem Lower quartile - 71 - 8.5th # Lower extreme - 45 - 1st # Upper extreme - 100 - 32th # Median - 78 - 18.5th # Upper quartile - 87.5 - 24.5th # What percentile is the score of 70? 70 is the 7th lowest of the 32 scores 7/32 =.21875 = 21.875% 22%

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Interquartile Range MaxMax MinMin Median Box-and-Whisker Plots show 5 important values from the data set. Lower extreme - lowest value Upper extreme - highest value Median - middle value Lower quartile - 25 th percentile value Upper quartile - 75 th percentile value Q3Q1 Box-Whisker Plot

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Box-Whisker Plot 70 8090 100 60 45 Interquartile Range MaxMi nQ3Q2Median 78 100 87.571 45 Paper Grade Problem Make a Box-and-Whisker Plot of

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Mean Absolute Deviation Set of data: 72, 85, 87, 89, 90, 93 xixi 938677 908644 898633 878611 85861 7286-1414 the sum of the differences between each entry in a sample and the mean of that sample is always equal to 0

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Variance Variance: A measure of dispersion that uses the squares of the deviations from the mean and gives greatest weight to scores farthest from the mean. Definition: The variance, v, of a set of data is the average of the squares of the deviation from the Mean.

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Variance – Model Problem Example: on 5 test scores, Fred earned grades of 92, 86, 95, 84, and 78. Find the variance. 1.Write # in order 2.Find mean 3.Find differences 4.Square differences 5.Apply formula Example: on 5 test scores, Fred earned grades of 78, 84, 86, 92, and 95. Find the variance. xixi 7887 8487 8687 9287 9587 -9 -3 5 8 81 9 1 25 64

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Standard Deviation Definition: the standard deviation,, of a set of data is equal to the square root of the variance. Result is in terms of original data, not the square of the values. Most important and widely used measure of dispersion in the world.

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Model Problem Example: on 5 test scores, Fred earned grades of 78, 84, 86, 92, and 95. Find standard deviation.

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Calculator and Model Problem STAT EDIT 1STAT CALC 1 ENTER

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. z-score Definition: the z-score is the number of standard deviations that a value is from the mean Example: A set of values has a mean of 85 and a standard deviation of 6. Find the z-score of the value 76.

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Aim: Measures of Dispersion Course: Alg. 2 & Trig. Our Favorite Model Problem A teacher marked a set of 32 papers. The grades were as follows: 90, 85, 74, 86, 65, 62, 100, 95, 77, 82, 50, 83, 77, 93, 72, 98, 66, 45, 73, 100, 50, 89, 78, 70, 75, 95, 80, 78, 83, 81, 72, 75. What value has a z-score of approximately 1.25?

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Aim: Measures of Dispersion Course: Alg. 2 & Trig.

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