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Unit 16: Statistics Sections 16AB Central Tendency/Measures of Spread Example 1:The following figures represent the average marks of some the grade 12 students applying to university next year. 75, 72, 76, 63, 64, 66, 70, 82, 76, 70 The data can be analyzed using measures of “Central Tendency”. They are the mean, median, and mode The mode is the data item or the class of data that occurs most often. It is possible for there to be more than one mode (bimodal, trimodal etc.). If all the data values or classes occur only once, then there is no mode. If we arrange our data we have 63, 64, 66, 70, 70, 72, 75, 76, 76, 82 The modes are 70 and 76 (bimodal) The median is found by arranging all the data in order from least to greatest and selecting the middle item. If there are an even number of data items, then there are two middle items. The median is the mean of these two items. 63, 64, 66, 70, 70, 72, 75, 76, 76, 82 The median for our data is 71 The mean is the measure of central tendency that is used most often. It is found by adding the numeric data and the result is divided by the number of data items. If an entire POPULATION is known, then we can calculate the mean of the population “” If we only have a SAMPLE of the population, then we calculate the mean of the sample “” If a set of data has outcomes x1, x2, …xn, then Mean x x i n 714 x 71.4 10 Many of these statistics can be determined using your calculator Measures of spread or variability is another way to compare data Percentile Ranking: If a student is in the 75th percentile, it means that the student finished higher than 75% of all the other students For example, if scores on a test were as follows: 63, 64, 66, 70, 70, 72, 75, 76, 76, 82 To calculate the 75th percentile 75% of 10 scores = 7.5 scores The 75th percentile is the 8th score from the bottom Therefore the 75th percentile score is 76 Quartiles When data is arranged in order from least to greatest, the median is the middle number in the set of data. Quartiles represent the data items that are one quarter and three quarters of the way through a set of data For example, consider the set of data 63, 64, 66, 70, 70, 72, 75, 76, 76, 82 Lower quartile Q1 Median Upper quartile Q3 66 71 76 Interquartile range Q3-Q1=76-66=10 Variance The average of the squares of the deviation from the mean is called the Variance. 1 ( xi x ) 2 n 2 This is a difficult formula to use So we can rearrange it to give. 1 2 xi x 2 n 2 Standard Deviation The Standard Deviation is the square root of the variance. It represents the “Average distance from the Mean” 1 2 2 x x i n If we refer back to our calculator HOMEWORK: PAGE 468 # 1, 2, 4 – 8 Correction#2a mean = 19.5, med = 15, mode = 15 PAGE 471 # 1 – 5 Do 16A#1 and 16B # 1 together