Central Tendency Growing Knowing com 2020 1 Central

Central Tendency Growing. Knowing. com © 2020 1

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. Growing. Knowing. com © 2020 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 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 Growing. Knowing. com © 2011 3

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 = 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. Growing. Knowing. com © 2011 4

Using Excel �Click the fx function button on the menu: The mean for the data in cells B 71 to B 75 is -1. 2 Growing. Knowing. com © 2011 5

Formula �Population Mean: �Sample Mean: �μ is called mu, the population mean. �x 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. Growing. Knowing. com © 2020 6

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. Growing. Knowing. com © 2011 7

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 = 3 + 4 + 8 = 15 �Multiply for the total: 3 x 20 + 4 x 10 + 8 x 5 = 140 �Mean = total sum / count � = 140/ 15 = 9. 3333 �Calculate the weighted mean for salary from the last slide � 1 x $800, 000 + 50, 000 x $35, 000 = 2, 550. 000, 000 /50001 = $50, 998. 98 Growing. Knowing. com © 2011 8

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. Growing. Knowing. com © 2011 9

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 + 4 / 2 = 3. Median is 3. Growing. Knowing. com © 2011 10

Median �Excel function: =MEDIAN(A 1: A 6) �The =MEDIAN function does not require you sort data Growing. Knowing. com © 2011 11

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 Growing. Knowing. com © 2011 12

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 Growing. Knowing. com © 2011 13

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, yellow, 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. Growing. Knowing. com © 2011 14

Excel �Use the =MODE. SNGL(a 1: a 4) function �#N/A is Excel saying ‘no mode’ �N/A means Not Available so no mode. �TIP: =MODE shows only the 1 st mode �For more than one mode �Highlight 4 or 5 rows of empty cells �Type =Mode. Mult(a 1: a 9) to select data �Don’t type Enter. Instead type CTRL+Shift+Enter keys together. �To make changes, delete the mode. mult and start over as editing is difficult. Growing. Knowing. com © 2020 15

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, extremeus meantremes. Growing. Knowing. com © 2011 Dr. Terry James 16