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Central tendency People 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 © 20112

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 data What is the mean salary where I work? What is the mean time it takes to drive to work? There are many types of mean calculations Arithmetic, weighted, harmonic mean, geometric mean, … We will learn arithmetic mean and weighted mean GrowingKnowing.com © 20113

Manually calculate the mean? Example: you are given numbers 1,2, and 3. Add each number to get a total (i.e. sum, symbol ∑) = 6 Count how many numbers you were given (symbol n) 3 Divide the total by the count 6 / 3 = 2 The mean includes every value, and finds the middle point of those values, in our example, number 2. GrowingKnowing.com © 20114

Using Excel Click the f x function button on the menu: GrowingKnowing.com © The mean for the data in cells B71 to B75 is -1.2

Formula Population Mean: Sample Mean: μ is called mu, the population mean. x̄ 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. GrowingKnowing.com © 20116

Beware of mean mistakes There are times when the mean can be misleading The average American has ½ a uterus and 1 testicle Almost 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. GrowingKnowing.com © 20117

Weighted Mean Calculate 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 = = 15 Multiply for the total: 3 x x x 5 = 140 Mean = total sum / count = 140/ 15 = Calculate the weighted mean for salary from the last slide 1 x $800,000, ,000 x $35,000 = 2, ,000 /50001 = $50, GrowingKnowing.com © 20118

Median The 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. GrowingKnowing.com © 20119

Median examples Odd count of data items If we are given 1, 4, 9, 5, 3 Sort: 1, 3, 4, 5, 9. Median is middle number = 4 You can calculate the middle number with (N + 1)/2 With 5 numbers. (5+1) / 2 = 3 so 3 rd number is 4. Even count of data items Sort numbers Find data at position N/2 and average with the data item above it in the sorted list Given: 1, 9, 4, 2, 2, 6 Sort: 1, 2, 2, 4, 6, 9 Take mean middle 2 numbers / 2 = 3. Median is 3. GrowingKnowing.com ©

Median Excel function: =MEDIAN(A1: A6) The =MEDIAN function does not require you sort data GrowingKnowing.com ©

Median versus Mean Mean for 1, 2, 3 is 2 Median for 1, 2, 3 is 2 Mean 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 GrowingKnowing.com ©

Avoiding median mistakes The 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 Excel Excel will automate any needed steps to avoid silly errors GrowingKnowing.com ©

Mode Mode is the data value that occurs most often We 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 often 1,2,2,3,3 has 2 modes, called bimodal. 2 and 3 occur the most 1,2,2,3,3,4,4 has 3 modes. 2,3,4 occur most often 1,2,2,3,3,4,4,4,5,5 has 4 as the mode. GrowingKnowing.com ©

Excel Use the =MODE(a1:a4) function #N/A is Excel saying ‘no mode’ N/A means Not Available TIP: =MODE shows only the 1 st 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 GrowingKnowing.com ©

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 GrowingKnowing.com ©