Mode

The mode is the most frequently occurring value in the distribution. This value will correspond with the peak of the histogram for the distribution. You can also scan through a frequency distribution to find the value that corresponds to the largest frequency or count.

The mode is the most frequently occurring value

For some distributions, there can be more than one pronounced peak in the distribution. We call these multimodal distributions. Check out figure 6.5 and try to determine why there are two modes.

Figure 6.5. A multimodal distribution

These multiple peaks are usually a result of having two distributions combined into one. In the case of figure 6.5, there are two modes for height because the distribution of heights for women and for men have been combined. Thus, there is a mode for women and a mode for men. We’ll discuss how statistical models can represent the impact of various predictor variables (e.g., sex) and a dependent variable (e.g., height) in future modules.

Although the mode can nicely correspond with the center of a distribution and is very easy to determine, it does not utilize very much information from the distribution. Rather than telling us about the various values in the distribution, it only tells us about one value. There is a silver lining here. The mode only requires us to count values, which means that we can indicate the mean of any variable. We can represent the mode of popular dog names (nominal scale), of stress ratings during final exams week (ordinal scale), of bank account balances (interval scale), and of the number of minutes of exercise a student gets each day (ratio scale).

Median

The median is often a more useful measure of central tendency when working with ordinal, interval, or ratio data because it Incorporated the ranking of the values rather than just the count or frequency of the values. As you may recall from earlier modules, the median is the value at which 50% of values are below and 50% of values are above. That is, if we order the values from smallest to largest and count halfway through the values, the median will be at the dividing point.

The median is the value at half-way point in ordered data

The location of the median is at \(\frac{n + 1}{2}\) where n is the number of values. If the median is at the halfway point, you may be wondering why the median is not at \(\frac{n}{2}\). Let’s find the median of the following 10 ordered values (number of yawns per student during a 30 minute lecture) to find out.

2 3 5 5 6 8 10 11 13 15

According to our formula, the median is at \[\frac{n+1}{2} = \frac{10 + 1}{2} = \frac{11}{2} = 5.5.\]

Of course, there isn’t a value at 5.5 so we will have to go halfway between the value at location 5 and location 6. To do that, we add the two values and divide by 2.

\[ \frac{\text{Location Below} + \text{Location Above}}{2} = \frac{6 + 8}{2} = \frac{14}{2} = 7 \]

Let’s rewrite our ordered values and insert the median

2 3 5 5 6 7 8 10 11 13 15

Notice how there are five values below the median and five values above the median. Because we have 10 values, 5/10 = .50 = 50% above and below the median. If we would have used \(\frac{n}{2}\), we would have chosen the value at 5 (which is 6), leaving 4 values (40%) below the median and 5 values (50%) above the median. By definition, using \(\frac{n}{2}\) will not yield the median.

Finding the mean of an odd number of ordered values is even easier because there is a value that divides the data into two equal parts. Let’s add another value to our list to see this.

2 3 5 5 6 8 10 11 13 15 29

Our list now has 11 values so or median is located at \(\frac{11 + 1}{2} = \frac{12}{2} = 6\). Counting from the left six values we find 8.

2 3 5 5 6 8 10 11 13 15 29

Once again, the data are divided in half with 5 values below and five values above the median.

Let us compare the median and mode of our data. The mode is five and the median is 8. Which value best represents the center of the data? Which value is more informative? Because the media incorporates the relative position of the values, we would use this value as a better representation of central tendency than the mode. However, this does not mean that the mode is not informative. It still tells us which value occurred most often. Soon, we will compare all our values of central tendency to help us determine how skewed or normal a distribution is.

The median utilizes information (i.e., order) of values but leaves out the values themselves. The last measure of central tendency incorporates the values of each datum in finding the balancing point in the data.

Mean

Reviewing quickly, the median is the value at the 50% mark or 50th percentile and the mode is the most frequently occurring value. The mean is the value at which the values below the mean balance out the values above the mean. It is calculated by taking the arithmetic mean or average of all the values.

Let us work with our current set of values:

2 3 5 5 6 8 10 11 13 15 29

The mean is calculated as

\[ \overline{x} = M=\frac{\sum_{i = 1}^{n}x_{i}}{n} = \frac{x_{1} + x_{2} + \ldots + x_{n}}{n} =\frac{2 + 3 + 5 + 5 + 6 + 8 + 10 + 11 + 13 + 15 + 29}{11}=\frac{107}{11}=9.73 \]

A few notes on the symbols you see. \(\overline{x}\) is pronounced “x bar” and is often used to represent the sample mean. Other times a capital “M” is used to represent a sample mean. ?? is the Greek letter “sigma.” This symbol is used to indicate that you will be adding together a series of numbers to the right of the symbol. In the extended form shown above, there is a starting location value and an ending value for the data. The starting value is below sigma (i = 1) that tells us to start with the first value. The ending value is above sigma (n is a representation of sample size) that tells us to add all the values until we reach the last value. Each value in the dataset can be represented as xi where i is the location of the value in our series of numbers. Table 6.2 shows the correspondence between xi values and the numbers in our series.

Table 6.2

Matching data values to xi representation

Xi	Value
X1	2
X2	3
X3	5
X4	5
X5	6
X6	8
X7	10
X8	11
X9	13
X10	15
X11	29

We will have more practice with this sigma notation shortly when we discuss standard deviation. In short, just remember that sigma notation tells you what to add up.

Now let us get back to the mean. Recall that we can conceptualize the mean as a balancing point for our data. Let’s build out this balancing metaphor by imagining that each value is connected to the mean with a metal rod such that the length of the rod is the numerical distance between the value and the mean. We’ll pair up values so that the most extreme (2 and 29) are connected to the mean followed by the next extreme (3 and 15) and so forth. Because we have an odd number, one score (10) will not have a counter balancing point.

Figure 6.5. Visualizing the mean as a balancing point

Although we may have fewer values on the right side of the scale, there is a much longer bar that compensates for the fewer and shorter bars compared to the left side. As such, the mean is the point at which the sum of the distances between the values and the mean balances out (i.e., equal to zero).

Rather than the graphic representation of distances, let us calculate the actual distance between the values and the mean.

Table 6.3

Balancing distances above and below the mean.

Xi	Xi - M
2	2-9.73 = -7.73	Distances below the mean add to -29.38
3	3-9.73 = -6.73
5	5-9.73 = -4.73
5	5-9.73 = -4.73
6	6-9.73 = -3.73
8	8-9.73 = -1.73
10	10-9.73 = 0.27	Distances above the mean add to 29.35
11	11-9.73 = 1.27
13	13-9.73 = 3.27
15	15-9.73 = 5.27
29	29-9.73 = 19.27
	\(\sum_{}^{}{(x_{i} - M)} = -0.03\)

The sum of the distances between the values below the mean and the distances between values above the mean cancel each other out (the -0.03 is due to rounding error). We can now be sure that our calculated mean is indeed the balancing point for our data.

The mean is a very informative measure of central tendency because it incorporates the value of each datum in the calculation. There are restrictions as to the data for which we can calculate the mean. We can only calculate the mean for data that have equal spacing between values on the scale on which it is measured. That is, we can only calculate the mean for interval and ratio scales of measurement.

Let us review the three measures of central tendency discussed thus far.

Table 6.4

Comparison of three measures of central tendency

Measure of Central Tendency	Definition	Determination	Scale of Measurement Requirement
Mode	Most frequently occurring value	Find value that appears most in data set (e.g., at peak of histogram or bar chart)	Nominal, Ordinal, Interval, Ratio
Median	Value at which 50% of data are above / below	Order values from smallest to largest, find value at \(\frac{n + 1}{2}\)	Ordinal, Interval, Ratio
Mean	Balancing point for data	Calculate average by adding up all values and dividing by the number of values (n) \[ \frac{\Sigma{X_i}}{n}\]	Interval, Ratio