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Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 113

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4.3 Measures of Variation Variation: Describes how widely data are spread out about the center of a distribution. ????How would you expect the variation to differ between the running times of theatre movies compared to running times for television sitcoms???? Theatre movie times more variation Television sitcoms less variation usually 30 or 60 minutes “Top of the Muffin to you!” ?????

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4.3 Measures of Variation How do we investigate variation? Study all of the raw data… Range… Quartiles… Five-number summary (BOXPLOT or BOX-and-WHISKER) … Interquartile range… Semi-quartile range… Percentiles… MAD… Variance & Standard Deviation…

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4.3 Measures of Variation Today: Semi-quartile range… Percentiles… MAD… Variance & Standard Deviation… 65 th Percentile! MAD???

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4.3 Measures of Variation Semi-quartile range The semi-quartile range is another measure of spread. It is calculated as one half the difference between the Upper Quartile (often called Q3) and the Lower Quartile (Q1). The formula for semi-quartile range is: (Q3–Q1) ÷ 2. Since half the values in a distribution lie between Q3 and Q1, the semi-quartile range is one-half the distance needed to cover half the values. In a symmetric distribution, an interval stretching from one semi-quartile range below the median to one semi-quartile above the median will contain one-half of the values. However, this will not be true for a skewed distribution.skewed distribution The semi-quartile range is not affected by higher values, so it is a good measure of spread to use for skewed distributions, but it is rarely used for data sets that have normal distributions. In the case of a data set with a normal distribution, the standard deviation is used instead. We will discuss standard deviation later.

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4.3 Measures of Variation EXAMPLE: Find the Semi-quartile range of the data. Semi-quartile = 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0 Q1 = 5.6 Q3 = 8.5 Semi-quartile = (8.5 – 5.6) 2 = 1.45 (Q3–Q1) ÷ 2

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4.3 Measures of Variation Percentiles The nth percentile of a data set is (an estimate) of a value separating the bottom values from the top (100 – n)%. A data value that lies between two percentiles is often said to lie in the lower percentile. You can approximate the percentile of any data value with the following formulas:

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4.3 Measures of Variation EXAMPLE: Percentiles. What percentile is the lowest score, Q1, Q2, Q3, and highest score? 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, 11.0 Lowest number = 0 percentile Q1 = 5.6 = 25 th percentile Q2 = ( )/2 = 7.45 = 50 th percentile Q3 = 8.5 = 75 th percentile Highest number = 100 th percentile

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4.3 Measures of Variation EXAMPLE: Percentiles. What percentile is the 9.3? 4.1, 5.2, 5.6, 6.2, 7.2, 7.7, 7.7, 8.5, 9.3, is the 9 th number out of ten, after the numbers are set in ascending order. Therefore, it is larger than 9 out of ten numbers, or the 90 th percentile. Note: One quartile is equivalent to 25 percentile while 1 decile is equal to 10 percentile and 1 quintile is equal to 20 percentile Think about it: P25 = Q1, P50 = D5 = Q2 = median value, P75 = Q3, P100 = D10 = Q4, P10 = D1, P20 = D2, P30 = D3, P40 = D4, P60 = D6, P70 = D7, P80 = D8, P90 = D9

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4.3 Measures of Variation MAD: Mean Absolute Deviation MAD is the mean of the absolute differences between the “sample mean” and the data values.

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4.3 Measures of Variation Example: Find the MAD (Mean Absolute Deviation). DATA: 10, 1, 3, 3, 3, 4, 5, 6, 5, 10 Mean = 5 ∑ 5, 4, 2, 2, 2, 1, 0, 1, 0, 5 = 22 MAD = 22/10 = 2.2

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4.3 Measures of Variation Variance… The variance of a random variable is a measure of statistical dispersion/distribution found by averaging the squared distance of its possible values from the mean. Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out. The unit of variance is the square of the unit of the original variable.

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4.3 Measures of Variation Example: Find the Variance. DATA: 10, 1, 3, 3, 3, 4, 5, 6, 5, 10 Mean = 5 ∑ 25, 16, 4, 4, 4, 1, 0, 1, 0, 25 Variance (s 2 ) = 80/9 = 8 8/9 ≈ 8.89

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4.3 Measures of Variation Standard Deviation… Universally accepted as the best measure of statistical dispersion/distribution. Standard deviation is developed because there is a problem with variances. Recall that the deviations were squared. That means that the units were also squared. To get the units back the same as the original data values, the square root must be taken.

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4.3 Measures of Variation Example: Find the Standard Deviation DATA: 10, 1, 3, 3, 3, 4, 5, 6, 5, 10 Mean = 5 ∑ 25, 16, 4, 4, 4, 1, 0, 1, 0, 25 Variance (s 2 ) = 80/9 = 8 8/9 ≈ 8.89 Standard Deviation = = ≈ 2.981

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4.3 Measures of Variation The Range Rule of Thumb… The is approximately related to the range of distribution by the following: We can use this rule of thumb to estimate the low and high values: low value ≈ mean – 2 × standard deviation high value ≈ mean + 2 × standard deviation The range rule of thumb does not work well when low and high values are extreme outliers. Therefore, use careful judgment in deciding whether the range rule of thumb is applicable.

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4.3 Measures of Variation The Range Rule of Thumb… EXAMPLE: The mean score on the mathematics SAT for women is 496, and the standard of deviation is 108. Use the range rule of thumb to estimate the minimum and maximum scores for women on the mathematics SAT. low value ≈ mean – 2 × standard deviation = 496 – (2×108) = 280 minimum high value ≈ mean + 2 × standard deviation = (2×108) = 712 maximum Is this reasonable? Of course, scores below 280 and above 712 are unusual on SAT’s.

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4.3 Measures of Variation HOMEWORK: Pg 174 # 3 Pg 175 # 9, 10, and 14 Pg 176 # 24, pg 176 # all (Letters c, d only)

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