Unit 32 STATISTICS STATISTICS AND PROBABILITY Probability concerns
Unit 32 STATISTICS
STATISTICS AND PROBABILITY � Probability concerns the possible outcomes (results) of experiments � Sample space is the group of all possible outcomes � Statistics are the basis of an analysis of a sample of information gathered about an operation. � A Sample is information gathered on a part of the operation � Decisions are made based on that analysis. 2
PROBABILITY SAMPLE SPACE � If a coin is tossed, the sample space contains two possible outcomes, heads (H) or tails (T). ◦ Written {H, T} � If a die is rolled, the sample space of the number of dots on the upper face ◦ Written {1, 2, 3, 4, 5, 6} � If two coins are tossed, the sample space has four possible outcomes ◦ Written {HH, HT, TH, TT} 3
PROBABILITY � Probability P of an event E occurring is: Where n = number of occurrences and s = all possible outcomes � Probability P of an event E not occurring E’ is: � If the probability P that something will happen then 1 -P is the probability it will not happen. P(E′) = 1 – P 4
PROBABILITY EXAMPLES � � Find the probability that a 4 will result when one die is rolled. n = 1 and s = 6 Find the probability P of at least one tail when two coins are tossed n = 3 {HT, TH, TT} ways to get a T s = 4 {HT, TH, TT, HH} 5
INDEPENDENT EVENTS � Events are independent if the probability that the second event will occur is not affected by what happens to the first event. � If A and B are independent events then the probability that both A and B will occur is P(A and B) = P(A)×P(B) 6
INDEPENDENT EVENTS EXAMPLE �A bag contains 3 yellow marbles and 4 blue marbles. A marble is drawn, replaced another drawn. �. Find the probability that first one is yellow and the second one is blue. 7
MEASURES OF CENTRAL TENDENCY � Mean (average) = � Median is the middle number of a group that is arranged in order of size. � Mode is the value that has the greatest frequency. � Bimodal means there are two greatest values of equal frequency 8
FINDING MEASURES OF CENTRAL TENDENCY � � � Find the mean, median and mode: 40 37 37 65 22 80 72 Median = 22 37 37 40 65 72 80 = 40 the middle number Mode is 37, number with the greatest frequency 9
QUARTILES AND PERCENTILES � Quartiles (Q 1, Q 2 (median), Q 3) divide the items in a set of numbers into four equally sized parts. � Arrange numbers in order from lowest to highest. ◦ Q 1 is the median of the lower half. ◦ Q 2 is the median. ◦ Q 3 is the median of the upper half. � Percentiles are numbers that divide the data into 100 equal parts 10
PERCENTILE EXAMPLES � Given: 1 2 2 4 5 5 6 7 7 8 9 10 11 13 13 14 15 15 16 18 20 20 21 25 25 � Find the 60 th percentile. � There are 25 numbers, so the 60 th percentile or 11
FREQUENCY DISTRIBUTION �A frequency distribution is an arrangement of a large group of numbers where most values are repeated ◦ One line contains a list of possible values and a second line contains the number of times each value was observed in a particular time ◦ The values in the first line are divided into intervals and the data arranged in lists. 12
FREQUENCY DISTRIBUTION �A histogram is a bar graph whose bars touch each other � A frequency distribution can be graphed as a histogram ◦ Use the intervals on the horizontal axis ◦ Use the frequencies on the vertical axis � Draw a histogram of the frequency distribution below 13
HISTOGRAM EXAMPLE � Draw a histogram of the frequency distribution below 25 20 15 10 5 211 -215 216 -220 221 -225 226 -230 231 -235 Hours 14
MEASURES OF DATA DISTRIBUTION � Range is the distance between the lowest and highest number in in a sample. � Variance is used mainly to find the standard deviation because it is not in the same unit of measure as the original data. � Standard Deviation gives a measure of how much the numbers are spread out from the mean. 15
VARIANCE AND STANDARD DEVIATION Where x is a measurement and n is the total number of measurements. 16
STANDARD DEVIATION EXAMPLE � Find the variance and standard deviation for the following set of numbers: 2. 5, 4. 6, 3. 2, 5. 1, 2. 1, 7. 3, 4. 9 Variance = Standard deviation = = 3. 226 Ans =1. 796 Ans 17
PRACTICE PROBLEMS � Find the probability in the following problems. 1. Getting a “head” when a coin is tossed. 2. Drawing a blue marble from a bag containing 3 blue and 5 yellow. 3. Rolling a sum of 5 or less on a pair of die. 4. Not drawing a queen from a deck of cards. 18
PRACTICE PROBLEMS (Cont) 5. Determine the mean, median and mode of the following set of numbers: 12, 15, 42, 37, 14, 9, 25, 32, 30 6. Determine the variance and the standard deviation of the following set of numbers: 76, 55, 77, 72, 39, 46, 47, 61, 59, 74, 43 19
PROBLEM ANSWER KEY 1. 2. 3. 4. 5. 6. 1/2 3/8 11/36 12/13 mean = 24. 7 median = 27. 5 mode = 32 variance = 199. 6 standard deviation = 14. 13 20
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