## Measures of Central Tendency

 Getting Started General Instructions | Introduction to Your Study Descriptive Statistics Histograms | Scatter Plots | Central Tendency | Standard Deviation | Confidence Intervals Relating Variables Correlation Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on Central Tendency at level 2. Level 1 | Level 2 | Level 3 Next Topic Frequency Histograms | Standard Deviation

### Explanation

Calculating Measures of Central Tendency
Level 1 showed how the three different measures of central tendency are each suited to different types of data. We will now see how the different measures can give different results depending on the pattern of values in your data. We will also show you how to calculate the values of the three different measures.
• The mode is calculated by counting how often each value occurs. There is no formula, you just count the values and see which one occurs most often. That value is the mode;
• The median is found by first sorting your data in order and then finding the middle value. If you have an even number of data points, there won't be a value right in the middle, so you take the two middle values, add them together and divide the answer by 2;
• The mean is calculated using a formula which adds all the values together and then divides by the number of values there are.
Now we know how to calculate the values, we can see how each different measure performs with different patterns of data.
• The mean is badly affected by extreme values. Imagine you had the data: 1,2,3,4,400. The mean is 410/5 = 82, which doesn't really reflect where any of the data lies
• The mode can be misleading if several values all appear equally often. It is useless if all the values in your data are different, as they each appear once only!
• The median can smooth out extreme values but can produce the least frequently occurring value, as in this example: 1,1,1,2,3,3,3

### Exploration

Here is the formula for calculating the mean of a set of numbers. Hover over any part of the formula for an explanation of what it signifies.

The whole formula means 'The mean of x (where x is any numeric variable, such as no. apps) is the sum of its values divided by the number of values there are.'

### Application

Here is the data from your study:
CostNo. Apps
715
66362
980110
13812
86183
14541
49346
54875
25123
1024100
43541
77275
7110
66359
38162
13815
86178
14559
49356
54856
25127
1009110
43546
772110
49346
54872
25123
988100
43541
66359
Calculator
( You need to enable Java to see this applet. )
Help
The Mean
Now we will calculate the mean of cost.
1. Use the calculator to add up all the values in the cost column.
Enter that value here:
2. Count the number of values in the data (n). Enter that value here:
3. Now divide the sum from step 1 by n. Enter that value here using 2 decimal places:
4. Now repeat the process for the no. apps column and enter the mean here:
To calculate the median and the mode, it is much easier to sort your data into order first.
Here is your data sorted with the position of each value noted along the top row.
 Position Cost No. Apps 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 71 71 138 138 145 145 251 251 251 381 435 435 435 493 493 493 548 548 548 663 663 663 772 772 861 861 980 988 1009 1024 5 10 12 15 23 23 27 41 41 41 46 46 46 56 56 59 59 59 62 62 72 75 75 78 83 100 100 110 110 110

The Mode
Now we will calculate the mode, if we can, for cost.
Find the mode by counting how many times each value appears and noting which appears most often. Also check to see if more than one value appears equally most often.

1. Let's see if it makes sense to calculate a mode for this data. The mode is the value that appears more than any other value. Is there a single mode for your data?
The Median
The median is in the middle of the ordered list of values.
1. Looking at cost, how many values are there?
2. Is there an exact midway point, or does the middle span two values?
3. At what position is the mid point? Enter its position or the first position if it spans two.
4. Find the value in the cost row at that position. If the mid point spans two places, add the numbers at those two positions together and divide what you get by two. The number you get is the median of cost. Enter it to 2 decimal places:
5. Now enter the median of no. apps. Use 2 decimal places