-3: Measure of Central Tendency Chapter : Descriptive Statistics The value at the center or middle of a data set. It is a tool for analyzing data. Part 1: Basic concepts of Measures of Center Ex. Data 17, 19, 1, 18, 0, 18, 19, 0, 0, 1 Mean (Arithmetic Mean): Found by adding the data values and dividing the total by the number of data. Sample mean x = n Population mean µ = Median: The middle of values when the original data values are arranged in order of increasing (or decreasing). Mode: The value that occurs with the greatest frequency. Midrange: The midway between the maximum and minimum values in the original data set. Midrange = maximum value + minimum value {Round-off Rule: Carry one more decimal place than is present in the original set of values.}
Part : Beyond the basics of Measures of Center Weighted Mean (ex. GPA): x = (w x) w = T otal of (Credit Quality P oint) T otal of Credits Ex. Mary took four classes in last semester, and her grades were A (4 credits), B (3 credits), A (3 credits), C (3 credits). Each letter grade has quality points as follows: A = 4, B = 3, C =, D = 1, F = 0. Compute her grade point average. Mean from a Frequency Distribution: (f x) x = f Ex. Find the mean of the data summarized in the given frequency distribution. Age of Best Actress When Oscar Was Won Frequency (f) Midpoint (x) f x 0 9 7 4.5 30 39 34 34.5 40 49 13 44.5 50 59 54.5 60 69 4 64.5 70 79 1 74.5 80 89 1 84.5 Totals 8
-4 : Measures of Variation Those tools show the characteristic of data s variation. Part 1: Basic Concepts of Variation Range =(maximum data entry ) - (minimum data entry ) Deviation: The difference between the entry (value) and the mean of the data. Variance: The average of the squares of the distance each value is from the mean. Standard Deviation: The square root of the variance. A.M ean V ariance Standard Deviation Sample x = n s = (x x) n 1 s = (x x) n 1 Population µ = σ = (x µ) σ = (x µ) Shortcut formula Sample Variance s = n (x ) ( ) n(n 1) Sample Standard Deviation n (x ) ( ) s = n(n 1) {Round-off Rule for Measure of Center and Variation}: Carry one more decimal place than is present in the original set of values. Round only the final answer, not values in the middle of a calculation.
Part : Beyond the Basic Concepts of Variation Range Rule of Thumb: Minimum usual value = mean standard deviation Maximum usual value = mean + standard deviation Empirical Rule for Data with a Bell-Shaped Distribution. 1) About 68% of all data values fall within 1 standard deviation of the mean. ) About 95% of all data values fall within standard deviation of the mean. 3) About 99.7% of all data values fall within 3 standard deviation of the mean. Chebyshev s Theorem: An general inequality applied to any distribution, including the empirical rule. 1) The proportion (or fraction) of any set of data lying within k standard deviations of the mean is always at least 1 1, where k is any positive number greater k than 1. ) For k =, at least 3 4 the mean. 3) For k = 3, at least 8 9 the mean. (or 75%) of all values lie within standard deviations of (or 89%) of all values lie within 3 standard deviations of
-5 : Measures of Position Boxplot (box-whisker diagram) Percentiles: The position measures used in educational and health-related fields to indicate the position of an individual in a group. Percentile of value x = (number of values less than x) + 0.5 total number of values 100 Finding Median = P 50 of the data. Ex. Data 34, 36, 39, 43, 51, 53, 6, 63, 73 Ex. Data 34, 36, 39, 43, 51, 53, 6, 63, 73, 79 5-umber Summary and Boxplot 1. Minimum data value. First quartile (Q 1 )= P 5 : At least 5% of the sorted values are less than or equal to Q 1, and at least 75% of the values are greater than or equal to Q 1. r total number of values = 0.5 3. Second quartile (Q )= P 50 : Same as the median; separates the bottom 50% of the sorted values from the top 50%. r total number of values = 0.50 4. Third quartile (Q 3 )= P 75 : At least 75% of the sorted values are less than or equal to Q 3, and at least 5% of the values are greater than or equal to Q 3. r total number of values = 0.75 5. Maximum data value.
Finding 5-umber Summary and constructing Boxplot Ex. 34, 36, 39, 43, 51, 53, 6, 63, 73, 79 Minimum data value = 34 P 5 = Q 1 r 10 = 0.5 r =.5 the 3 rd value 39 P 50 = Q r 10 = 0.50 r = 5 the value between 5 th and 6 th 51 + 53 = 5 P 75 = Q 3 r 10 = 0.75 Maximum data value = 79 r = 7.5 the 8 th value 63 Interquartile Range (IQR) : (Q 3 Q 1 ) Outliers with IQR Lower fence: Q 1 1.5 (IQR) Upper fence: Q 3 + 1.5 (IQR)
z-scores (Standard score): The z-score measures how far each data value is from the mean of a data set using Standard Deviation as a yardstick. z-score z = x x s : Sample z-score z = x µ σ : Population Ordinary values: z-score Unusual values: z-score < or z-score > If the z-score is positive, the score is above the mean. If the z-score is 0, the score is the same as the mean. And if the z-score is negative, the score is below the mean. {Round-off Rule for z-scores}: Round z-score to two decimal places. Ex: Find the z-score for each test and state which is higher. Test A: x = 38, x = 40, s = 5 Test B: x = 94, x = 100, s = 10 Ex: Determine whether the score of Test C is unusual or not. Test C: x = 98, x = 7, s = 1