Measures of Center. Statistics

​Measures of Central Tendency

Concepts:

​The average of the sales from a sample of representatives is a statistic, and the average of sales obtained from the entire population is a parameter.

​Before starting...

​When calculating, NEVER ROUND UP until the final answer is calculated.

That is to avoid loosing significant value with each calculation.

​Mean:

​Sample Mean
(statistic)

Population Mean
(parameter)

Multiple Choice

What is the value of the mean?

• 61.4

• 64.8

• 54.8

• 57.9

​Mean for grouped data

​In case of grouped data, the calculation of the mean is achieved by adding the product between the frequency with the midpoint, divided by the total of data.

Mean for grouped data

​Mean in Google Sheets

​Step 1

​Mean in Google Sheets

​Step 2

​SUMMARY

The Mean

1. The mean is found by using all the values of the data.

2. The mean varies less than the median or mode when samples are taken from the same population and all three measures are computed for these samples.

3. The mean is used in computing other statistics, such as the variance.

4. The mean for the data set is unique and not necessarily one of the data values

​SUMMARY

The Mean

5. The mean cannot be computed for the data in a frequency distribution that has an open-ended class.

6. The mean is affected by extremely high or low values, called outliers, and may not be the appropriate average to use in these situations.

​Median:

​Median for Grouped Data:

​l = lower bound median class
cs = class size
n = total amount of data
f = frequency
cf = cumulative frequency 

Median in Grouped Data (example):

Find the Median with the following information:

Median in Grouped Data (example):

Build the frequency table:

Median in Grouped Data (example):

Build the frequency table:

​n = 51
Median class = 52/2 = 26th
[145, 150)

​Median in Google Sheets

​SUMMARY

The Median

1. The median is used to find the center or middle value of a data set.

2. The median is used when it is necessary to find out whether the data values fall into the upper half or lower half of the distribution..

3. The median is used for an open-ended distribution.

4. The median is affected less than the mean by extremely high or extremely low values.

​Mode

​Mode in Google Sheets:

​SUMMARY

The Mode

1. The mode is used when the most typical case is desired.

2. The mode is the easiest average to compute.

3. The mode can be used when the data are nominal or categorical, such as religious preference, gender, or political affiliation.

4. The mode is not always unique. A data set can have more than one mode, or the mode may not exist for a data set.

​Weighted Mean:

It is the type of mean that considers an additional factor, and it is used when the values are not all equally represented.

​Weighted Mean (Example):

​Weighted Mean Google Sheets:

Calculate the average time you spend exercising four days a week over a month or four weeks. The time you spent exercising on any given day is the data set. The number of days you exercised for an average time is the weight you'll use.

• 7 days you exercised for 20 minutes

• 3 days you exercised for 45 minutes

• 4 days you exercised for 15 minutes

• 2 days you were supposed to exercise and did not

​Weighted Mean Google Sheets:

​Midrange

​Midrange (Example)

​SUMMARY

The Midrange

1. The midrange is easy to compute

2. The midrange gives the midpoint

3. The midrange is affected by extremely high or low values in a data set.

​DISTRIBUTION SHAPES AND MCT

​DISTRIBUTION SHAPES AND MCT

​DISTRIBUTION SHAPES AND MCT

​SUMMARY

Multiple Choice

Second, assume you work for the teachers’ union and want a raise for the teachers. Use the best measure of central tendency to support your position.

• Mode

• Mean

• Median

• Any of them would work

Multiple Choice

Teacher Salaries

The following data from several years ago represent salaries (in dollars) from a school district in Greenwood, South Carolina.

10,000 11,000 11,000 12,500 14,300 17,500

18,000 16,600 19,200 21,560 16,400 107,000

First, assume you work for the school board in Greenwood and do not wish to raise taxes to increase salaries. Compute the mean, median, and mode, and decide which one would best support your position to not raise salaries.

• Mode

• Median

• Mean

• All of them work

Open Ended

Explain how outliers can be used to support one or the other position

Multiple Choice

If the salaries represented every teacher in the school district, would the averages be parameters or statistics?

• Parameters

• Statistic

Multiple Choice

Which measure of central tendency can be misleading when a data set contains outliers?

• MEAN

• MODE

• MEDIAN

• ALL OF THEM

Multiple Choice

When you are comparing the measures of central tendency, does the distribution display any skewness? Explain.

• It doesn't display any skewness

• It presents a skewness to the right

• It presents a skewness to the left

• It presents a U shape

Airport Parking The number of short-term parking spaces at 15 airports is shown. Find the mean, median, mode, and midrange for the data.