We next examine the idea of measuring central tendencies. A measure of central tendency is simply a way to express a single representative number for a set of numbers. It is a characteristic value for a group of values. Inside this concept are things we think about every day: averages, variation, ranges. We will be examining three measures of central tendency:
NOTE: many vCalc equations are embedded throughout vCalc descriptive pages like this page. Even though they may not stand out in the text, if you hover over the name of an equation it will likely be linked to an actual, pop-up executable equation. For example: Arithmetic Mode
The arithmetic mean (`barX`), or mean for short, is the result of adding a set of values and dividing that sum by the number of values that were added. The bar annotation above the X just means that this is the mean of X. The mean (`barX`) is what you in everyday life refer to as the average: the average height of a group of people, the average time to run a marathon, the average cost of a bar of soap. All these averages are calculated the same way we calculate the mean (`barX`) here.
Click here to try the arithmetic mean yourself.
The Greek symbol `sum` is used to represent the sum of a set of things.
The formula then for the mean is written:
`barX = (sum_1^N(X_i))/N`
You can read this mathematical expression as:
The arithmetic mean is the sum of the values of the set `X` from `X_1` to `X_N`, where `N` is the count of the values in the set `X`.
The subscript or index `i` on the `X` represents the index in the sequence of members of the set `X` from 1 through `N`. For example, `X_1` is the first number in the set `X`, `X_2` is the second number in the set `X`, `X_3` is the third number in the set `X`, ... ,`X_i` is the i-th number in the set `X`, and `X_N` is the n-th number (or last number) in the set `X`.
If Joe knew the ages of his coworkers, he could find the mean or average age of employees that he works with.
Let's say that Sally is 23, Bill is 32, John is 19, Fred is 32, Reese is 25 and Joe himself is 30.
To compute the mean age of Joe and his coworkers you would first sum their ages. The set of ages in years, `X`, is:
23, 32,19, 32, 25, 30
and the number of ages in the set `X` is `N` = 6
The sum of those ages is:
`sum_1^6(X_i) = 23 + 32 +19 + 32 + 25 + 30 = 161`
The mean then is computed by simply dividing the sum of the ages (161) by the number `N`
`barX = (sum_1^6(X_i))/N = 161/6 = ~28.83` 1
You can try this equation in vCalc yourself. Just copy the comma-separated list of ages (23, 32,19, 32, 25, 30 ) and paste them into the vCalc equation for the arithmetic mean: Copy the list above and Click Here
The mean is the average. If you you are comparing test scores and the scores are pretty much evenly distributed across the whole range of possible test scores -- let's say between 0 and 100, you expect the mean to be right in the middle, so around 50. But most people are trying to pass the test, so there tends to be more scores in most normal cases above 60 than there is below 60. That means the average will turn out to be something closer to 70 if everybody taking the test has been studying.
So, where the calculated mean lies in the range of values indicates a "tendency" to be above or below the center of a a distribution of values/
The mode is the value that appears most often in a set of data. So we count up how many times each particular value might appear in a set of values and the value that occurs the most times in the set is the mode.
The Mode calculation indicates what part of the range of values in a set of numbers occurs most often.
If you think about a set with a large portion of its values close to one part of the range, you can see the mean (`barX`) will be skewed toward the Mode. The mean will be closer to the mode than it is to the center of the range. So, there is a relationship between the mean and the mode/
We will see later that in a normal distribution, the mean, median and mode all coincide/
Let's use the same set of ages from the example we examined for the mean above:
`X` = 23, 32,19, 32, 25, 30
We find that one of the age values exists in the set of coworker ages more than any other value. Bill and Fred are both 32 and there are no other coworkers that are the same age. So, the mode of `X` is 32.
Try copy/pasting this example data set into the Try the Mode calculation
Wherever the Mode lies in the range of values, there is some amount of clustering of values/ This most common value will have a "tendency" to pull the average toward it.
The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. Values in the data set above the Median are all larger values than the values of the data below the Median
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If we use our coworker ages data set from above: `X` = 23, 32,19, 32, 25, 30 , where `N` = 6 is the number of values in the data set
We can find the median by ordering the set: `X+"(ordered)"` = 19, 23, 25, 30, 32, 32
We calculate half the values is N * 0.5 = 3. However since we have an even number of values in this data set, the middle of the data set is between the 3rd element from the bottom and the 3rd element from the top. So, we find the average between these two values.
Thus the Median =`(25+30)/2` = 27.5
Try it for yourself with the vCalc Median equation. Copy the comma-separated data from our example and paste the data into the Median equation HERE/
The Median, like the Mode and the Mean, indicates a "tendency" to be near the center of the range of values/ The Median/wiki/Median show you what value lies at the very center of the range/ So, a Median that is a low value in the range tells you that values will be clustered tighter in the low values of the range.
See Also