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 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

5. 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}

6. 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)

7. 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.

8. 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

9. FINDING MEASURES OF CENTRAL TENDENCY
Find the mean, median and mode:
Median = = 40 the middle number
Mode is 37, number with the greatest frequency

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. PERCENTILE EXAMPLES
Given:
Find the 60th percentile.
There are 25 numbers, so the 60th percentile or

12. 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.

13. 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

14. HISTOGRAM EXAMPLE Draw a histogram of the frequency distribution below25 20 15 10 5
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. 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 =
= 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 = standard deviation =