OPTIMA Statistics

Subsection 1.3.1 The Mode

We have already seen the term mode used in the context of a graph. Recall that the mode of a histogram or bar graph is the class or category that has the tallest bar. That is, it is the class or category that appears the most in the data set. This seems to be a reasonable way to describe a “typical” element of a data set and leads to the following definition.

The mode is unique among our measures of center as it is the only one that applies to both quantitative variables and qualitative variables. Consider the following examples.

While the mode is an easy way to measure the center, and is in fact the only way we have to measure the center of a set of qualitative data, it is usually not the best choice for measuring the center of a set of quantitative data. To see why this may be so, consider the next example.

In summary, you should use the mode to measure the center of any qualitative set of data, or as a quick, but not definitive, measure of center for quantitative data. But be careful! The mode may not exist, there may be multiple modes, and if there is a single mode it may be very different from the “typical” values in the data set.

Subsection 1.3.2 Midrange and Range

When we think of the “typical” value in a set of data, we may also think of the word middle. Our next measure of center is exactly that—the middle of the data set.

Notice that the midrange does involve some computation using numerical values, so it can only be used with quantitative variables, where such computations make sense. To measure how spread out the data is from the midrange, we use the range.

Let's try out these new definitions with an example.

The midrange and range are useful tools for measuring the center and spread of a set of data, but there are several problems with them. The biggest problem is that both the midrange and range are extremely sensitive to outliers.

To see how an outlier can affect the midrange and range, let's take another look at the fish-length data from the previous example.

Notice what a big difference the single 6.5 inch fish made! This is what we mean when we say that the midrange is sensitive to outliers. Another problem with the range and midrange is that they only use the extreme maximum and minimum values of the data set. Ideally, we would want every value in the set of data to have some part in computing the “typical” value and the “spread” of the data. For this reason, we will keep exploring different measures of center and spread.

Subsection 1.3.3 Median and Quartiles

To reinforce why we need a measure of center that is resistant to outliers, consider the following example.

Once again, the outliers in this data set are affecting the midrange. To solve that problem, we introduce a new measure of center that is both resistant to outliers (meaning that they do not affect it very much) and considers more than just the two extreme values in the data set.

You may have heard of the median before. It is often used as a measure of center in data sets where we expect outliers, such as housing prices. Another common set of data in which the median is used is household income. The phrase “median household income” may sound familiar. For those of us working in Washington State with Bill Gates, it is a good thing that the median is used to measure typical incomes! Let's practice computing the median.

Notice that the 1000 in the second data set is a definite outlier. However, in computing the median, the actual value does not matter. The only thing that matters is its relationship to the rest of the data. This 1000 could be replaced by a more reasonable 220 and the median would not change. Bill Gates could make only $200,000 a year, and the median household income in Washington State would not change! This is what it means to be resistant to outliers.

To measure the variation in a set of data, we can use a slight twist on the median. Notice that the median divides the set of data into two pieces. If we take each of those pieces and divide them in half, we are dividing the data into quarters. We can then look at the range between the first quarter of the data and the third quarter of the data. This gives us a measurement similar to the range. But since we are only looking at the middle half of the data, we will not be affected by any outliers at the extremes.

The range between \(Q_1\) and \(Q_3\) is of particular interest and, as mentioned above, it has a special name.

To see how this works, let's revisit the fish-length example.

Subsection 1.3.4 Mean and Standard Deviation

While the median is a good measure of center to use in the case where we know a set of data contains outliers, it does have one major disadvantage. Namely, it does not take the value of every data point into consideration. In many instances, we want each value in the data set to have weight in determining the “typical” value. To achieve this goal, we turn back to an already familiar measure of center, commonly called the average. We will, however, use a more precise name for this measure.

Notice that there is a different symbol used for the mean depending on whether it is the mean of an entire population, or just the mean of a sample. For a population, the Greek letter \(\mu\) (pronounced “myoo”) is used for the mean. For a sample, we use \(\overline{x}\) (read “x bar”) to represent the mean. Keep this distinction in mind, as it can be a clue as to what type of data you are dealing with on a homework or exam problem.

Another thing to note is that the name arithmetic mean is important. First, we don't call this the average, even though most people think of it as the average. The reason is simply that the word average means “typical” and we have seen several ways to measure the “typical” value in a data set. Next, we specify that this is the arithmetic mean because there are actually different types of means. We could, for example, use the geometric mean, which is given by the formula:

For many data sets the geometric mean and arithmetic mean are very close. But you wouldn't want to have your homework average computed using the geometric mean. To understand why not, think about what would happen if only one of your assignments was not turned in! In this text we will always use the arithmetic mean. For that reason, we can simplify things by stating up front that we mean the arithmetic mean when we say mean.

The measure of spread most commonly used with the mean is called the variance. This can be thought of as the average (or mean) distance that each data point lies away from the mean. If we compute the average, however, we would get zero because of the +/- signs. So we square these differences, resulting in the following definition.

Again we use a different symbol to represent the variance of a population, the Greek letter \(\sigma^2\) (pronounced “sig-muh squared”), and a sample, the Latin letter \(s^2\) (pronounced “s squared”). But the variance of a population also differs in a more substantial way from that of a sample. Notice that in a population, we divide the sum of the differences squared by \(n\text{,}\) the number of values in the population. But in a sample, we divide by \(n-1\text{.}\) This is an important distinction to remember. The reason for it is, unfortunately, beyond the scope of the course, but involves the fact that in a sample there is less variation than in the entire population.

One problem with using the variance as a measure of spread is that the units are different from the units of the data set. For example, if we say that the mean number of children in a US household is 2.2 with a variance of 0.4, then the mean is in “children”. That is, there are an average of 2.2 children in US households. The variance, however, has units “square children.” What does it mean to say that the number of children varies by 0.4 square children? To remedy that, we often take the square root of the variance to get a measure of spread that is in the same units as the mean.

Let's test out these new formulas on an example.

The number for this example turned out very nice. This is not always the case. Usually the standard deviation will involve decimals. In general, you should not round your answers during any intermediate steps, and then round your standard deviation to one decimal more than the number of decimals in the original data set. One quick way to estimate the standard deviation is based on the range of the data.

In the above example, our estimate of \(\frac{8-3}{4} = 1.25\) is not terribly good, but this can be a quick and easy way to approximate the standard deviation if all you have are the minimum and maximum values of your population data.

Subsection 1.3.5 Selecting Appropriate Summaries

We have now seen four different sets of numerical summaries:

• the mode,

• the midrange and range,

• the median and IQR, and

• the mean and standard deviation,.

But how do we know when to use a particular measure of center? To better answer this question, let's review the strengths and weaknesses of each of these numerical summaries in tabular format.

As we can see, each method has its own strengths and weaknesses. Based on these, we can summarize when to use which measures of center and spread, as outlined in the observation below.

Often it is helpful to compute several summaries. One useful comparison relates the mean and median together. When the mean is larger than the median, the data will have a histogram that is skewed right. When the mean is smaller, the histogram will be skewed left. Finally, if the mean and median are equal, the histogram will be exactly symmetric.