Data Description Measures of Central Tendency

Chapter 2 showed how to organize raw data into frequency distributions and then present the data by using various graphs. This chapter will show the statistical methods that can be used to summarize the data. The most familiar of these methods is finding the average.

The average speed of a car crossing midtown Manhattan during the day is 5.3 mph

The average number of minutes an American father spends alone daily with his child is 42 minutes

“average” when you stop and think of it is a funny concept. Although it describes all of us it describes none of us…. While none of us wants to be the average American, we all want to know about him or her.

The vast majority of people have more than the average number of legs.

The average American man is five feet nine inches tall…

The average American woman is 5 feet 3.6 inches tall.

The average American is sick in bed seven days a year…

On the average day, 24 million people receive animal bites…

By his or her 70 th birthday, the average American will have eaten 14 steers, 1050 chickens, 3.5 lambs, and 25.2 hogs…

The word average is ambiguous, since several different methods can be used to obtain an average.

Loosely stated, average means the center of the distribution or the most typical case.

Even if a shoe store owner knows that the average man’s foot is a size ten, they wouldn’t be in business very long is the only size they stocked was size ten.

Measures of Central Tendency this is where the notes begin…

Statisticians use samples taken from populations; however, when populations are small, it is not necessary to use samples since the entire population can be used to gain information.

Measures found by using all the data in a population are called parameters

Measures obtained by using data values of samples are called statistics

General Rounding Rule DO NOT ROUND UNTIL THE FINAL STEP!!!!

Mean Arithmetic average

Mean is found by adding the values of the data and dividing by the total number of values. The symbol for Mean is (pronounced: x bar)

Formula for sample Mean

Formula for population Mean

In statistics, Greek letters are used to denote parameters, and Roman letters are used to denote statistics

In most cases the Mean is not an actual data value 2, 3, 4, 8, 10 mean = 5.4

Rounding Rule for the Mean The mean should be rounded to one more decimal place than occurs in the raw data. For example, if the raw data are given in whole numbers, the mean should be rounded to nearest tenth. If the data were given in tenths, then the mean should be rounded to hundredths, and so on.

Finding the mean for grouped data Abbreviation for frequency – f Abbreviation for midpoint - x m

Finding the mean for grouped data First set up your headers

Fill in the data

find the midpoint of each class

multiply the frequency by the midpoint for each class

find the sum of column D

Divide the sum of column D by n to get the grouped mean

Is this mean the mean of the data?

The Median

The symbol for median is MD

The median is the halfway point in a data set. Before you can find this point, the data must be arranged in order. When the data set is ordered, it is called a data array.

Steps in computing the median 1.Arrange the data in order 2.Select the middle point

The number of rooms in the seven hotels in downtown Pittsburgh is 713, 300, 618, 595, 311, 401, 292. Find the median. 292, 300, 311, 401, 595, 618, 713 median

The number of tornadoes that have occurred in the US over an 8 year period follows. Find the median median

Since the median falls between two numbers, you have to do a little math Add the two numbers on either side and divide by two.

If there is an odd number of datum the median is the middle data value If here is an even number of datum the median is the average of the middle pair of data values.

The Mode

The mode is the value that occurs most often in a data set A data set may have one mode, more than one mode, or no mode at all.

The following data represent the duration (in days) of Space Shuttle voyages for the years Find the mode

The mode is 8

If every datum is unique (no number repeated) the data set is said to have no mode. Do not say mode is zero. This indicates that there is a mode and that it is 0.

The mode for grouped data is the modal class. The class with the largest frequency

Example 3-14

mean = 20,000 median = 12,000 mode = 9,000 What is the average?

Midrange

Symbol for midrange is MR defined as the sum of the lowest value and the highest value in a data set, divided by 2 a very rough estimate of the middle

Weighted Mean Sometimes, one must find the mean of a data set in which not all values are equally represented.

Consider the case of finding the average cost of gasoline for three taxis. Taxi 1 buys /gal Taxi 2 buys /gal Taxi 3 buys /gal What was the average cost of gasoline?

Taxi 1 buys /gal Taxi 2 buys /gal Taxi 3 buys /gal

NO!

Assignment Exercise set 3-2 page