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

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

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

Range - lowest to highest amount 78 100 45 71 87.5 Quartiles Range - lowest to highest amount 78 100 45 71 87.5 Q1 Median Q3 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. Find the lower and upper quartile of Paper Grade Problem n = 32 Lower quartile is the median of the first 16 numbers Average of the 8th & 9th numbers (70 + 72)/2 = 71 Upper quartile is the median of the last 16 numbers Average of the 24th & 25th numbers (86 + 89)/2 = 87.5

What percentile is the score of 70? Percentiles Q1 Median Q3 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 What percentile is the score of 70? Lower quartile - 71 - 8.5th # Lower extreme - 45 - 1st # Upper extreme - 100 - 32th # Median - 78 - 18.5th # Upper quartile - 87.5 - 24.5th # 70 is the 7th lowest of the 32 scores 7/32 = .21875 = 21.875%  22%

Box-and-Whisker Plots show 5 important values from the data set. Box-Whisker Plot Q3 Q1 Interquartile Range Max Mi n 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 - 25th percentile value Upper quartile - 75th percentile value

Make a Box-and-Whisker Plot of Box-Whisker Plot Interquartile Range Mi n Q2 Median Q3 Max Paper Grade Problem Make a Box-and-Whisker Plot of 70 80 90 100 60 45 45 71 78 87.5 100

Mean Absolute Deviation Set of data: 72, 85, 87, 89, 90, 93 xi 93 86 7 90 4 89 3 87 1 85 -1 72 -14 14 the sum of the differences between each entry in a sample and the mean of that sample is always equal to 0

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.

Variance – Model Problem Example: on 5 test scores, Fred earned grades of 92, 86, 95, 84, and 78. Find the variance. Example: on 5 test scores, Fred earned grades of 78, 84, 86, 92, and 95. Find the variance. Write # in order Find mean Find differences Square differences Apply formula xi 78 87 84 86 92 95 -9 81 -3 9 -1 1 5 25 8 64

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.

Model Problem Example: on 5 test scores, Fred earned grades of 78, 84, 86, 92, and 95. Find standard deviation.

Calculator and Model Problem STAT  EDIT  1 STAT  CALC  1 ENTER

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.

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?