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Unit 32

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.

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Unit 32

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  1. Unit 32 STATISTICS

  2. 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.

  3. 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}

  4. Probability P of an event E occurring is: Where n= number of occurrences and s = all possible outcomes Probability P of an event Enot 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 PROBABILITY

  5. 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} PROBABILITY EXAMPLES

  6. 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) INDEPENDENT EVENTS

  7. A bag contains 3 yellow marbles and 4 blue marbles. A marble is drawn, replaced and another drawn. .Find the probability that first one is yellow and the second one is blue. INDEPENDENT EVENTS EXAMPLE

  8. 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 MEASURES OF CENTRAL TENDENCY

  9. 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 FINDING MEASURES OF CENTRAL TENDENCY

  10. QUARTILES AND PERCENTILES • Quartiles (Q1, Q2 (median), Q3) divide the items in a set of numbers into four equally sized parts. • Arrange numbers in order from lowest to highest. • Q1 is the median of the lower half. • Q2 is the median. • Q3 is the median of the upper half. • Percentiles are numbers that divide the data into 100 equal parts

  11. 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 60th percentile. There are 25 numbers, so the 60th percentile or PERCENTILE EXAMPLES

  12. 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. FREQUENCY DISTRIBUTION

  13. 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 FREQUENCY DISTRIBUTION

  14. Draw a histogram of the frequency distribution below 25 20 15 10 5 211-215 216-220 221-225 226-230 231-235 HISTOGRAM EXAMPLE Hours

  15. 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.

  16. VARIANCE AND STANDARD DEVIATION • Where x is a measurement and n is the total number of measurements.

  17. 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 EXAMPLE = 3.226 Ans • Standard deviation = =1.796 Ans

  18. PRACTICE PROBLEMS • Find the probability in the following problems. • Getting a “head” when a coin is tossed. • Drawing a blue marble from a bag containing 3 blue and 5 yellow. • Rolling a sum of 5 or less on a pair of die. • Not drawing a queen from a deck of cards.

  19. PRACTICE PROBLEMS (Cont) • Determine the mean, median and mode of the following set of numbers: 12,15,42,37,14,9,25,32,32,30 • Determine the variance and the standard deviation of the following set of numbers: 76,55,77,72,39,46,47,61,59,74,43

  20. PROBLEM ANSWER KEY • 1/2 • 3/8 • 11/36 • 12/13 • mean = 24.7 median = 27.5 mode = 32 • variance = 199.6 standard deviation = 14.13

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