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What Is Central Tendency?

Central tendency is defined as the typical value of a set of data. There are three main ways that we measure central tendency: mean, median, and mode.

The mean is the average of all the data. To find the mean, add up all the values and then divide by the number of values. The median is the value that is exactly in the middle of a list of the data, after it's been arranged in numerical order. The mode is the most commonly occurring number in the data set. Although these measures of central tendency are very useful, sometimes it's not enough just to know the mean, median, or mode. Sometimes, you need to represent data graphically in order to see any underlying patterns.

Central Tendency Dot Plots

One of the simplest ways to represent a set of data graphically is a dot plot. Dot plots are graphs that show the number of times that each data point occurs by the placement of dots. For example, if a teacher wanted to know how many books his students read last month, he could collect the data, organize it into a table like the one on your screen below, and then make a dot plot like the one shown on screen below that. In this dot plot, each dot represents one student.

This dot plot represents how many books were read by students in a class during one month. Each dot represents one student.

Dot plots are really good for identifying the mode of a data set, which in this case are seven and nine. This means that more students read seven or nine books during the last month than any other number of books.

The median can be found by locating the point that is exactly in the middle. Here, the median is 4.5 books because there are 50 dots in total, and dot 25 is at four books and dot 26 is at 5 books.

To find the mean, you need to calculate a weighted average or write out the data. Multiply each number of books by the number of students above it. Then add all these and divide by the total number of students (50) to find the mean.

Mean = ((0 x 3) + (1 x 3) + (2 x 7) + (3 x 6) + (4 x 6) + (5 x 3) + (6 x 4) + (7 x 8) + (8 x 2) + (9 x 8)) / 50 = 4.8 books

Write out the data: 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9

If I were to put this data in Desmos, I would type into a line:

L= [0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9]

If I wanted Desmos to display a dotplot for me, I would type into a new line: dotplot (L)

If I wanted Desmos to calculate the following items below for me, I would type into a new line:

a) mean: mean(L)

b) median: median(L)

*CHALLENGE: what if I wanted Desmos to calculate the range for the data set?

-What is range? range = max-min

-What would I type in Desmos? stats(L)

Central Tendency Histograms

A histogram shows the frequency of how often a certain value(s) or a range of values occur. Histograms are used to visualize the distribution of data.

To make a histogram:

Decide how many groups you want to have, and separate the data into groups.

On the x-axis, label the value of each group. The y-axis will always represent the frequency.

Draw a bar that is the height of the frequency for each group.

A histogram of our example data is shown on your screen below. This histogram has a group size of 1 book, so there are 10 different groups.

From a histogram, you can quickly see how the data is distributed, and this helps you decide how to analyze the data further.

If I wanted Desmos to display a histogram for me, I would first type in my data set as L = [ ... ].

Then, in a new line, I would type histogram (L).

Central Tendency Box Plots

Box plots, also known as box-and-whisker plots, only show a summary of the data, which is known as the five number summary.

The five-number summary tells us 5 points: minimum, Q1, median, Q3, maximum.

They are used to quickly compare different sets of data with each other.

*In order to get the 5 number summary in Desmos, I would first type in my data set as L = [ ... ].

Then, in a new line, I would type: stats(L)

To make a box plot:

First find the median of the data and draw a line at that point.

Next, divide the data into four quartiles. With the data arranged in sequential order, the lowest 25% would be in the first, or lower, quartile, the second 25% in the second quartile, the third 25% in the third quartile, and the highest 25% in the fourth, or upper, quartile.

Draw lines on the graph showing the locations of the lower and upper quartile and draw a box enclosing these and the median. This box will contain the middle 50% of the data, a region known as the interquartile range.

Finally, add bars to the left and right of the box to show the maximum and minimum values of the data in the set.

The box plot on your screen shows our original example data in blue. On the box plot, you can clearly see the median of the data set, the interquartile range, and the maximum and minimum extreme values.

Box plots can also give you information about the amount of spread in the data. If the box is very small, you know that the data values are generaally close together with little variation, but if it is large, then the data is more spread out and there could be a possible outlier.

One big advantage of box plots is that they can allow you to compare different sets of data quickly. If the teacher wanted to compare the books read by more than one class (Class A and Class B), he could show this on the same box plot and compare the data.

If I am referring to percentages, a box plot as a whole represents 100% and we typically break it up into 1/4 or 25% units using the 5 number summary.

The interquartile range is the middle 50% of the data and we calculate this by Q3-Q1.

Next, we will review the appropriate times to use a specific measure of center to describe the data.

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