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? $$ \begin{array}{lllllll} 41, & 53, & 38, & 32, & 115, & 47, & 50 \end{array} $$

Short answer

The mean (average) of the data set is 53.71, the median is 47, and there is one detected outlier, which is 115.

Step-by-step solution

Calculate the Mean

To obtain the mean \(\bar{x}\), we add all numbers together and divide that sum by the number of data points: \( \frac{41 + 53 + 38 + 32 + 115 + 47 + 50}{7} = \frac{376}{7} = 53.71 \)

Calculate the Median

To calculate the median, we first need to arrange the numbers in ascending order: \(32, 38, 41, 47, 50, 53, 115\). Since there are 7 numbers, the median is the number in the middle, which is 47 in this case.

Identify Outliers

Outliers are data points significantly different from others. They are usually defined as values that are 1.5xIQR away from the first or the third quartile. Here, we don't have values of quartiles given, so we will use a simpler but less precise definition: a value is considered an outlier if it is far away from the median, the mean or both. Looking at the ordered data, the number 115 stands out because it is much higher than the others. So, 115 is an outlier.

Key concepts

Mean Calculation

Understanding the mean, or average, of a data set is essential for analyzing data in a meaningful way. To calculate the mean, we sum all the numbers in the set and then divide that sum by the total count of numbers.
In the context of our exercise, adding up all the numbers in the set $$ 41 + 53 + 38 + 32 + 115 + 47 + 50 = 376, $$
and then divided by the number of data points, which is 7, yields $$ \frac{376}{7} = 53.71. $$
The mean is particularly useful as it gives us a central value around which other numbers tend to cluster. It's important to note, however, that the mean can be significantly affected by outliers - extreme values that differ greatly from the other numbers in the data set. Outliers can inflate or deflate the mean, which sometimes may give a misleading representation of the overall data set.

Median Calculation

The median is another form of a central tendency measure that represents the middle value of a data set when it's arranged in ascending order. Unlike the mean which utilizes all values, the median is solely determined by the dataset’s positional context. When dealing with an odd count of data points, the median is simply the middle value. For an even number of points, it is calculated as the average of the two middle values.
For the data set provided in the exercise, $$ 32, 38, 41, 47, 50, 53, 115, $$
we identify the median by counting to the central value of the ordered list, which is 47. Since there are three values on either side of 47, it stands as the median. The median is robust against the influence of outliers and thus provides a better central tendency measure for skewed distributions or when outlier data points are involved.

Outliers Identification

Identifying outliers – those data points that are significantly different from the rest of the data – is crucial for a more accurate statistical analysis. While there are sophisticated methods involving the interquartile range (IQR), a simpler approach is to look for values that are notably far from both the mean and median.
In our exercise's data set, the number 115 is an outlier, as it lies away from the cluster of other data points and skews the mean. This exceptional value can affect the interpretation and statistical calculations of the data set. Recognizing and potentially excluding outliers, when justified, can lead to more representative analytics, especially in mean calculations. However, it is equally important to consider the context of the data, as outliers may carry significant meaning or insights, like indicating an error or highlighting a rare event in a particular field of study.