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.

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}

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

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}

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)

INDEPENDENT EVENTS EXAMPLE 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.

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

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

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

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

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.

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

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

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.

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

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 = = 3.226 Ans Standard deviation = =1.796 Ans

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.

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

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