How do we find averages for data in the information we are testing? This is where central tendency becomes useful! The average can usually be found in the middle of the bulk of the data, the center. How do you find these central tendencies? There are a few types, each with different uses. Read on.
*Note: this guide may use symbols you aren’t familiar with. View our complete list of Statistics Symbols here.
Measures of Central Tendency
The measures of central tendency are methods that are used to finding the center of the data. The type of data that you have in your problem, either quantitative or qualitative information, and normal or skewed distribution shapes, will help you decide which measure of central tendency you choose to describe the center of your data. Measures of central tendency include mean, median, and mode.
Mean
The mean, or calculated average, of a set of data is found by taking the sum of the data and dividing by the total number or samples. The mean of a population is called μ (mu) the Greek letter for “m.” The mean of a sample is called x̄ (x bar).
The formula for the mean is: x̄ = Σ xi /𝑛 where 𝑛 stands for the sample size, Σ represents taking the sum of the data, and xi represents each value that will be added.
If you were reading this formula in plain English, it would say “Add up all the data values, then divide by the sample size.”
The mean can be found in any set of data, but is best used to describe symmetric/bell- shaped/normal data. Skewed data will move the mean towards the skew. In symmetric data, it centralizes to describe the true center. So, for symmetric data, consider “mean” as your average.
Median
The median is much like the median of a street – it is in the middle of the numbers. In order to find the median, we first list the numbers in the sample from least to greatest, and then find the number in the center. If the set of data contains an odd number of items, then there will be one a number in the center, which is the median. If the set of data contains an even number of items, then there will not be two numbers in the center. In this case, you would add the two numbers in the center and then divide them by two to find the median. The median can be found in any set of data, but is best used to describe skewed data. It is resistant to skew and outliers. So, for skewed data, consider “median” as your average.
Mode
The mode is the data item that occurs the most often in a set of data. It repeats the most. It can be found in any set of data, but the mode is best used in data sets that contain only qualitative data (non-numerical) in order to find the measure of central tendency. So, for descriptive, qualitative data, consider “mode” as your average. If there is no number or value that occurs the most often, than there is no mode.
Tech Help
In StatCrunch
Step 1:
At the top, select “Stat,” then “Choose Summary Stat,” then “Select Columns”
Step 2:
Choose the column that contains your data by highlighting your choice Under “Statistics:” select mean, median, and mode by clicking each word on the left side
Step 3:
Click “Compute!”
In Excel (2 Methods)
First method
Step 1: In any Excel entry box, enter...
- =Average (select all numbers in your column by highlighting them)
- =Median (select all numbers in your column by highlighting them)
- =Mode (select all numbers in your column by highlighting them)
Second method
Step 1: Find measures all at once with “Data Analytics Add On”
Select “Data” tab at the top, Click “Data Analysis Add On,” then “Select Descriptive Statistics”
- Under Input Range, select all data/numbers with your mouse
- Check the Summary Statistics Box
- Click “OK”
This will automatically populate a new sheet in Excel with various descriptive statistics data including all the measures of central tendency.
Step 1: Find measures all at once with “Data Analytics Add On”
Select “Data” tab at the top, Click “Data Analysis Add On,” then “Select Descriptive Statistics”
- Under Input Range, select all data/numbers with your mouse
- Check the Summary Statistics Box
- Click “OK”
This will automatically populate a new sheet in Excel with various descriptive statistics data including all the measures of central tendency.
Now You Try It! Practice Problems
Example One
You decide to measure the average amount of likes you receive on Instagram posts. Suppose you have collected these five numbers below from your most recent posts. Find the Mean, Median, and Mode.
5, 17, 21, 95, 42
Example Two
While walking, you decide to observe how many people you pass on your regular walking trail at the local park, in order to find the average. You take note during 10 of your walks and find the numbers below. Find the measures of central tendency.
5, 5, 5, 3, 3, 3, 2, 2, 1, 1, 7
Example One
5, 17, 21, 95, 42
Mean: Add up all the numbers and divide by how many you have
Median:
First put the numbers in order from least to greatest: 5, 17, 21, 42, 95
Next, we eliminate numbers on the left and right until the middle is found.
Cancel one from the front and one from the back: (Eliminating 5 and 95) we would have 17, 21, 42
Repeat this process until there is at most two numbers left. (Eliminating 17 and 42) we would have 21. Our median is therefore 21.
Mode: No numbers repeat. There is no mode.
Example Two
5, 5, 5, 3, 3, 3, 2, 2, 1, 7
Mean:
Median:
First, we put them in order again. 1, 2, 2, 3, 3, 3, 5, 5, 5, 7
This time the numbers are even. So, we will have two middle values. We will still eliminate numbers on the left and right until we find the middle.
We can see that 1, 2, 2, 3, are on the left. 5, 5, 5, 7 are on the right. This leaves 3, and 3 in the middle
To find the true center, we divide the two middle numbers by 2.
3 is our median
Mode:
This time, there are two modes. The numbers 5 and 3 occur the same number of times, so, there are multiple modes.
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