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13.2: Measuring the Center and Variation of Data Kalene Mitchell Allie Wardrop Sam Warren Monica Williams Alexis Carroll Brittani Shearer Common Core Standards Summarize and describe distributions. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. CCSS.MATH.CONTENT.6.SP.B.5.C Grade 6 The Mean… Definition: The mean or average, of a collection of values is x̅ = S/n, where S is the sum of the values and n is the number of values. The symbol x̅ should read as “x bar.” Commonly referred to as: Average Arithmetic mean The Mean… Example: Find the mean of the following values: 13, 18, 13, 14, 13, 16, 14, 21, 13 Page 735 All 12 players on the Uni Hi basketball team played in their 78- to-65 win over Lincoln. Jon Highpockets, Uni Hi’s best player, scored 23 points in the game. How many points did each of the other players average? The Median… Let a collection of n data values be written in order of increasing or decreasing size. Largest # Smallest # Half values above & half values below Can be used in situations where we cannot get a proper measurement but can only rank data in order ex: Arrange workers in order of their performance. A worker in the middle would represent median performance. 50% of the workers do not work as hard 50% of the workers work harder The Median… If n is odd, the median, denoted by x, is the middle value in the list. X= middle Value If n is even, x^ is the average of the two middle values. The symbol x^ should be read as “x hat.” ^ X= n1 (first middle #) + n2 (second middle # / 2) The Median… Example: Find the median of the following values: 11, 13, 5, 19, 33, 12, 14, 15, 16, 11, 10, 32 If we take away one values, how would it change the problem? 11, 13, 5, 19, 33, 12, 14, 15, 16, 11, 10 The Mode… Definition: A value that occurs most frequently in a collection of values. If two values occur equally often and more frequently than all other values, there are two or more modes. (Bi….Tri…) Commonly referred to as: The number seen most often The Mode.. Example: Find the mode of the following values: 44, 35, 36, 43, 41, 40, 35, 37, 34, 36, 37, 33, 36 Measures of Variability Variability- how far spread out the scores or data points are There are four frequently used measures of variability Range: highest minus lowest score Interquartile range- range of middle 50% scores Standard deviation Variance Upper and Lower Quartiles Definition: Consider a set of data arranged in order of increasing size. Let the number of data values, n, be written as n=2r when n is even, or n=2r+1 when n is odd, for some integer r. The lower quartile, denoted by QL, is the median of the first r data values. The upper quartile, denoted by Qu, is the median of the last r data values. Page 737 For the given data set of this problem, determine the lower and upper quartile. A = (12,7,14,15,9,11,10,11,0,8,17,5) Interquartile Range IQR Definition: Difference between the upper and lower quartile IQR= Qu - QL In other words: 75th percentile – 25th percentile Outlier Definition: data value that is LESS THAN QL –(1.5 x IQR) or GREATER THAN +(1.5 x IQR) In other words, data points or scores that are atypical of the other values in the data set. So less than 25th percentile – (1.5 x 75th percentile) Or greater than 75th percentile – (1.5 x 25th percentile) Box Plot or Box-and-Whisker Plot Definition: consists of a central box extending from the lower to the upper quartile, with a line marking the median and with line segments, or whisker, extending outward from the box to the extremes. Standard Deviation Definition: Let X1, X2, X3…..Xn be the values in a set of data and let x denote their mean. What it means- How far from the normal Formula: How to Find Standard Deviation Step 1: Find the mean Step 2: Find the difference each number is from the mean Step 3:Take each difference and square it Step 4: Add those numbers together Step 5: Divide that sum by the total number of terms Step 6:Take the square root of that number
