Question

For each set of data (a) Find the mean \(\bar{x}\). (b) Find the median \(m\). (c) Indicate whether there appear to be any outliers. If so, what are they? $$ \begin{array}{l} 110, \quad 112, \quad 118, \quad 119, \quad 122, \quad 125, \quad 129, \\ 135, \quad 138, \quad 140 \end{array} $$

Short answer

The mean of the given data set is 124.8. The median is 123.5. There are no outliers in this data set.

Step-by-step solution

Compute the Mean

Add all numbers together and then divide by the count of numbers to get the mean of the data set. Mean \( \bar{x} = \frac{110+ 112+ 118+ 119+ 122+ 125+ 129+ 135+ 138+ 140}{10} = 124.8 \)

Compute the Median

First, sort the numbers in ascending order. Because there are an even number of numbers, the median is the mean of the two middle numbers. Median \( m = \frac{122 + 125}{2} = 123.5 \)

Identify Outliers

To find any outliers, first we need to compute the first quartile (Q1) and the third quartile (Q3). These are the median of the first half and the second half of the data set, respectively. Then we compute the interquartile range (IQR) by subtracting Q1 from Q3. Any number less than Q1 - 1.5 * IQR or greater than Q3 + 1.5 * IQR is considered an outlier. In our case, no data points fall below Q1 - 1.5 * IQR or above Q3 + 1.5 * IQR, therefore there are no outliers in our data set.

Key concepts

Mean Calculation

Understanding how to calculate the mean, often referred to as the average, is one of the foundational skills in descriptive statistics. To calculate the mean, you sum up all the values in a data set and then divide the total by the number of values you have. The mean represents the central value of the data set. For instance, given the numbers 110, 112, 118, 119, 122, 125, 129, 135, 138, and 140, you first add them up to get 1,248, and then divide this sum by 10, which is the count of numbers. This results in a mean of 124.8.

It's important to note that the mean can be highly sensitive to extreme values or outliers in the data set. This means that the presence of very high or very low values can significantly affect the mean, pulling it in the direction of the outlier.

Median Calculation

The median is the middle value of a data set when it has been ordered from least to greatest. It is a measure of central tendency that is less affected by outliers and skewed data. When working with an even number of data values, like we have in our set - 10 numbers, the median is calculated by taking the mean of the two middle numbers. In our example, once the values are ordered, the two middle numbers are 122 and 125. The median is, therefore, the average of these two numbers, which is \( \frac{122 + 125}{2} = 123.5 \).

When the number of values is odd, the median is simply the middle value. In a skewed distribution, where values are not evenly spread, the median provides a better sense of the 'typical' value than the mean.

Outlier Identification

An outlier is a data point that differs significantly from other observations in a data set. Identifying outliers is crucial because they can distort statistical analyses and lead to incorrect conclusions. Outliers can be identified using various methods, but one common technique is through the interquartile range (IQR). An outlier in this context is typically defined as a value that is more than 1.5 IQRs below the first quartile (Q1) or above the third quartile (Q3).

In the data set provided, after computing the IQR, we check each value to see if it falls outside the range of \( Q1 - 1.5 \times IQR \) and \( Q3 + 1.5 \times IQR \). If no values fall outside this range, as with our example, we determine that there are no outliers present.

Interquartile Range (IQR)

The interquartile range, or IQR, is a measure of statistical dispersion and is calculated as the difference between the third and first quartile (Q3 - Q1). The first quartile (Q1) is the median of the lower half of the data set, excluding the median, and the third quartile (Q3) is the median of the upper half. To find the IQR, we subtract the value of Q1 from Q3.

For instance, if we have the quartiles Q1 = 119 and Q3 = 135 from a data set, the IQR would be \( 135 - 119 = 16 \). The IQR denotes the range within which the central 50% of the data fall and is used to identify anomalies in our data. Specifically, any data points that lie 1.5 times the IQR above Q3 or below Q1 could be deemed outliers, as they fall far outside the range of what is considered 'usual' within the data set.