GrowingKnowing.com © 2011 1. Central tendency People like to know, what are the central values? You can use central values to measure how you are doing,

GrowingKnowing.com © 2011

1GrowingKnowing.com © 2011

Central tendencyPeople like to know, what are the central

values?You can use central values to measure how

you are doing, compare with others, or what to expect.

My grade is above the average. I am doing well.

The average shows this employer promotes quickly, so I will apply for a job.

GrowingKnowing.com © 2011 2

Mean Arithmetic mean is often called the “mean” or

average.The mean is the number in the middle;

add the size of every number for a total,then divide by the how many numbers given (count).

The mean is an excellent place to start analyzing your dataWhat is the mean salary where I work?What is the mean time it takes to drive to work?

There are many types of mean calculationsArithmetic, weighted, harmonic mean, geometric mean, …

We will learn arithmetic mean and weighted mean


Manually calculate the mean?Example: you are given numbers 1,2, and 3. Add each number to get a total (i.e. sum, symbol ∑)

1+2+3 = 6Count how many numbers you were given (symbol

n)3

Divide the total by the count6 / 3 = 2

The mean includes every value, and finds the middle point of those values, in our example, number 2.


Using Excel Click the fx function button on the menu:


The mean for the data in cells B71 to B75 is -1.2

FormulaPopulation Mean:

Sample Mean:

μ is called mu, the population mean. xE is pronounced "x bar“, the sample mean. Σ is Sigma, which is the sum of the data values. x is a variable representing each of the data values. N is the count of the data values for a population. n is the count of the data values for a sample.


Beware of mean mistakesThere are times when the mean can be misleadingThe average American has ½ a uterus and 1

testicleAlmost everyone earns less than the average

salary!One CEO earns 800 million, 50,000 workers earn

$35,000 Average salary of $50,999 50,000 people out of 50,001 earn less than average!

When data includes an extreme value, called an outlier, using the mean may be misleading.


Weighted MeanCalculate the mean score for people playing a game 3 people got a score of 20, 4 got 10, and 8 people got 5

Count how many numbers: n = 3 + 4 + 8 = 15Multiply for the total: 3 x 20 + 4 x 10 + 8 x 5 = 140Mean = total sum / count

= 140/ 15 = 9.3333Calculate the weighted mean for salary from the last

slide 1 x $800,000,000 + 50,000 x $35,000 = 2,550.000,000

/50001= $50,998.98


MedianThe median is the number in the middle using a

count of how many numbers we are given Given the data, 1,2, 999. Median = 2 which is the number in the middle.

Always sort the data first Sorting ensures the number in the middle does not

depend on the order numbers are given.For an odd list of numbers, take the middle

number.For even list of numbers, use the mean of middle

2 numbers.


Median examplesOdd count of data items

If we are given 1, 4, 9, 5, 3Sort: 1, 3, 4, 5, 9.Median is middle number = 4

You can calculate the middle number with (N + 1)/2With 5 numbers. (5+1) / 2 = 3 so 3rd number is 4.

Even count of data itemsSort numbersFind data at position N/2 and average with the data

item above it in the sorted listGiven: 1, 9, 4, 2, 2, 6Sort: 1, 2, 2, 4, 6, 9Take mean middle 2 numbers. 2 + 4 / 2 = 3. Median

is 3.GrowingKnowing.com © 2011 10

MedianExcel function: =MEDIAN(A1: A6) The =MEDIAN function does not require you

sort data


Median versus MeanMean for 1, 2, 3 is 2Median for 1, 2, 3 is 2Mean and median are close if the data has no

outliers.

Mean for 1, 2, 999 is 334.Median for 1, 2, 999 is 2.

Use median instead of mean if outliers are extreme


Avoiding median mistakesThe common error is forgetting to sort the

data.If the data list is short, students find medians

easy to find without a computer. To avoid errors, use =MEDIAN function if you

have access to ExcelExcel will automate any needed steps to avoid

silly errors


ModeMode is the data value that occurs most oftenWe are often interested in the mode

What is the most popular color car?Who is the popular leader?

Cars sold by color: blue, blue, yellow, black, black, black.Black is the mode.

You can have no mode or multiple modes.1,2,3,4 has no mode. No data value occurs most often1,2,2,3,3 has 2 modes, called bimodal. 2 and 3 occur

the most1,2,2,3,3,4,4 has 3 modes. 2,3,4 occur most often1,2,2,3,3,4,4,4,5,5 has 4 as the mode.


ExcelUse the =MODE(a1:a4) function

#N/A is Excel saying ‘no mode’ N/A means Not Available

TIP: =MODE shows only the 1st mode Excel 2010 has mode.sngl and

mode.mult=Mode.sngl works the same as =mode=Mode.mult handles multiple modes but

is awkward to use We suggest sorting data, then run

=MODE multiple times to scan for multiple modes


Can you write a poem about the mean? No means I am a man of means and a child of extremes. There is a mean between extremes; extremes predicting the means, if you get what I mean ?

That's extreme, an extreme mean, or did I mean, a mean extreme ? Ex-mean, is no longer mean. Such nonsense, extremeanus meantremes. Dr. Terry James


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GrowingKnowing.com © 2011 1. Central tendency People like to know, what are the central values? You can use central values to measure how you are doing,