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Measure of Central Tendency Measures of central tendency – used to organize and summarize data so that you can understand a set of data. There are three common measures: Mean Median Mode

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Mean Mean = sum of the data items total number of data items The symbol for the mean is Use the mean to describe the middle set of data that DOES NOT have an outlier. Outlier – a data value that is much higher or much lower than the other data values in the set. Often referred to as the average.

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Median Median – the middle value in the set when the numbers are arranged in order. If a set contains an even number of data items, the median is the mean of the two middle values. Use to describe the middle of a set of data that DOES have an outlier.

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Mode Mode – the data item that occurs the most times. Possible for a set of data to have no mode, one mode, or more than one mode. Use the mode when the data are nonnumeric or when choosing the most popular item.

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Range To calculate the range subtract the smallest data value from the largest data value. Example: 21, 15, 16, 25, 13, 18 Range = 21 – 13 = 8

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Examples Find the mean, median, and mode. Which measure of central tendency best describes the data. Weights of textbooks in ounces:12, 10, 9, 15, 16, 10 Mean = 12 + 10 + 9 + 15 + 16 + 10 = 72 = 12 6 6 Median: 9, 10, 10, 12, 15 16 = 10 + 12 = 22 = 11 2 2 Mode: 9, 10, 10, 12, 15 16 = 10 Since there is no outlier, the mean best describe the data Range = 16 – 9 = 7

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Mean Absolute Deviation (M A D) This measure averages the absolute values of the errors.

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List your data set (x)Find the mean of your data ( ) Find (x – )Find |(x – )| Find the sum of ∑|(x – )|. Divide your sum by the total of your sample size (n)… The mean absolute deviation is…. Math I Unit 4 Calculating the Mean Absolute Deviation (MAD)

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Box-and-whisker plot (Boxplot) A box-and-whisker plot is a visual way of showing median values for a set of data. The lower quartile (Q1) represents one quarter of the data from the left and the upper quartile represents three quarters (Q3) of the data from the left. The five important numbers in a box-and whiskers plot are the minimum and maximum values, the lower and upper quartiles, and the median. The interquartile range = Q3 – Q1

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How to construct Boxplot Steps: 1) Make a number line using equal interval (from the minimum to the maximum value) 2) Calculate the median Q2 3) Calculate Q1 = median of the first quarter 4) Calculate Q3 = median of the third quarter 5) Make a box connecting the quartile 6) Draw a line from the minimum value to Q1 and another from Q3 to the maximum value

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Example 4,6,8,10,12 Minimum value = 4 Median = 8 Q1 = (4 + 6) ÷ 2 = 5 Q3 = (10 + 12) ÷ 2 = 11 Maximum value = 12 Interquartile range = 11 – 5 = 6 Median( ) 4 6 8 10 12 X largest X smallest

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Shape of a Distribution Describes how data is distributed Measures of shape – Symmetric or skewed – See how mean compares to median, and possibly the mode Mean = Median(=Mode?) Mean < Median(< Mode?) ( Mode<?) Median < Mean Right-Skewed Left-SkewedSymmetric

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Distribution Shape and Box-and-Whisker Plot Right-Skewed (positive) Mean>median Left-Skewed (negative) Mean < median Symmetric (zero)

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