Introduction to Statistics

Queen Victoria, Cunard's newest cruise ship by savannahgrandfather

• Judith and Francine, both age 19, have decided to go on a Caribbean cruise, and they want to have an enjoyable time, which means that they want to travel with other people their own age. They buy tickets for a cruise where the average age of the other passengers is 20 years. Sounds like fun, no?

Why Study Statistics? • Two students in two different schools each

have marks of 95 percent. Which student should receive an award for getting the 'higher' mark?

• How do doctors decide that teenagers should or should not get hepatitis vaccine?

• Judith and Francine, both age 19, have decided to go on a Caribbean cruise, and they want to have an enjoyable time, which means that they want to travel with other people their own age. They buy tickets for a cruise where the average age of the other passengers is 20 years. Sounds like fun, no?

Why Study Statistics? • Two students in two different schools each

have marks of 95 percent. Which student should receive an award for getting the 'higher' mark?

• How do doctors decide that teenagers should or should not get hepatitis vaccine?

• Judith and Francine, both age 19, have decided to go on a Caribbean cruise, and they want to have an enjoyable time, which means that they want to travel with other people their own age. They buy tickets for a cruise where the average age of the other passengers is 20 years. Sounds like fun, no?

Why Study Statistics? • Two students in two different schools each

have marks of 95 percent. Which student should receive an award for getting the 'higher' mark?

• How do doctors decide that teenagers should or should not get hepatitis vaccine?

Can you imagine their surprise at the start of the cruise when they discover that all the other passengers are parents (average age 32) with children (average age 8)?

big_girl_04_m1_screen by pntphoto

Statistics: the branch of mathematics that deals with collecting, organizing, displaying, and analyzing data.

statistic: a number that describes one aspect of a group of data.EXAMPLE: mean, median, mode, range, standard deviation, etc...

datum: one bit (piece) of information.

data: many bits (pieces) of information.

Types of Dataquantitative data: data that is numeric(eg. height, weight, time..)

There are two kinds of quantitative data: continuous and discrete

continuous data: can be represented using real numbers (eg. height, weight, time, etc..)

discrete data: can be represented by using ONLY intergers (eg. # of people, # of cars, # of animals, etc..)

qualitative data: data that is non-numeric (eg. colours, flavours, etc...)

Measures of Central Tendency

mean: ( A.K.A. 'the arithmetic mean") the symbol for mean is "x bar". The arithmetic average of a set of values.

where x is the meanwhere Σx means the sum of all data (x) in the set (Σ is called "sigma")where n is the number of data in a set

EXAMPLE: find the average mark this set of 5 quizzes: 48,52,65,45,65.

Measures of Central Tendency

median (med): 1) the middle value in an ordered (from smallest to largest) set of data.2) if there are an even number of data, the median is the average of the middle pair in an ordered set of data.

EXAMPLE: find the median of these quiz scores: 12,10,17,11,15

SOLUTION: 10, 11, 12, 15, 17

12 is the median.

EXAMPLE: find the median of these scores: 12,10,17,11,15,11

SOLUTION: 10,11,11,12,15,17

the median is 11.5

mode (mo): the datum that occures most frequently in a set of data.

EXAMPLE: find the mode in the set of quiz scores: 12,10,17,11,15,11

SOLUTION: the mode is 11 because it occurs more often that any other number in the set.

Mean, Median, Mode, ...

A clerk in a men's clothing store keeps a weekly record of the number of pairs of pants sold. The following is her list for two weeks.

Calculate the mean, mode, and median for the data shown.

Mon Tue Wed Thur Fri Sat 34 40 36 36 38 38

32 36 36 42 34 34Week1Week 2

Bimodal Distribution

Measures of Dispersion (Variability)

Dave can drive to work using the downtown route or the perimeter route. The downtown route is shorter, but it has more traffic, and can become quite crowded. The driving times in minutes for each route (arranged in ascending order) for 5 days are shown on the table below.

Downtown Route 15 26 30 39 45Perimeter Route 29 30 31 32 33

The average driving time for each route is 31 minutes. Which route should he take?

Measures of Dispersion (Variability)determine how "spread out" or varied" a set of data is.

Range: the difference between the largest and smallest value in a set of data.

EXAMPLE: find the range of ages of people in our class

highest value: lowest value: RANGE:

with teacher MR K. ___ yrs old.highest value: lowest value:RANGE:

Measures of Dispersion (Variability)

Back to our example:Dave can drive to work using the downtown route or the perimeter route. The downtown route is shorter, but it has more traffic, and can become quite crowded. The driving times in minutes for each route (arranged in ascending order) for 5 days are shown on the table below.

Downtown Route 15 26 30 39 45Perimeter Route 29 30 31 32 33

Find the range associated with taking each route.

Downtown Route Perimeter Route

Measures of Dispersion (Variability)determine how "spread out" or varied" a set of data is.

Standard Deviation (σ): a measure that shows how the data are spread about the mean value. Every value in the data set is used in calculating the standard deviation.

Downtown Route 15 26 30 39 45Perimeter Route 29 30 31 32 33

Find the standard deviation associated with taking each route to Dave's work using your calculator.

Downtown Route Perimeter Route

Let's apply what we've learned ...

The mean math marks and standard deviation for two classes are shown below. Assume that 68 percent of the marks in each class are within one standard deviation of the mean mark.

(b) Bert in Class A and Beth in Class B each have a mark of 82%. How many standard deviations are they from their class means? Who appears to have the better mark?

(a) In which class is the set of marks more dispersed?

mean mark (μ) standard deviation (σ)Class A 74 4Class B 72 8

HOMEWORK

The following numbers represent the number of cars sold by Metro Motors in one week:

Monday Tuesday Wednesday Thursday Friday Saturday 4 5 8 9 7 9

1. Determine the following statistics:

(a) mean (b) mode (c) median (d) range

2. Which measure of central tendency may be the least significant? Explain.

HOMEWORK

The two sets of data show the weights of potatoes in bags. There are six bags in each set.

The mean weight of each set of bags is 50 pounds. Which set has the greater standard deviation? How do you know? (Do not do any calculations.)

Set #1 49 51 48 52 47 53Set #2 40 60 45 55 35 65

HOMEWORK

A class of 30 students received the following marks in a mathematics examination. Calculate the mean, median, range, and standard deviation.

78 92 62 52 65 5953 63 68 73 71 6369 74 73 81 55 7175 81 84 77 80 7541 57 91 62 65 49

HOMEWORK