Question

Find the mean, the median, and the mode of the collection of numbers. $$4,2,10,6,10,7,10$$

Short answer

The mean is 7, the median is 7, and the mode is 10.

Step-by-step solution

Calculate the Mean

To find the mean of the data set, add up all of the numbers and then divide by the total count of the numbers. This gives \((4 + 2 + 10 + 6 + 10 + 7 + 10) / 7 = 7\). So, the mean is 7.

Calculate the Median

To find the median, first sort the numbers in ascending order. The sorted list becomes: \(2, 4, 6, 7, 10, 10, 10\). There are 7 numbers and the median is the middle number, in this case, 7. Therefore, the median is 7.

Find the Mode

The mode is the number that appears most frequently. In this data set, the number 10 occurs thrice, which is more frequent than any other number, so the mode is 10.

Key concepts

Understanding the Mean

The mean, often referred to as the average, is a fundamental statistical measure that represents the central point of a data set. To calculate the mean, you simply add all the numbers in the set together and then divide by the quantity of numbers. It's a straightforward but powerful concept used to find the typical value.

For example, in the exercise provided, the numbers 4, 2, 10, 6, 10, 7, and 10 are added to make 49, and since there are 7 numbers in total, dividing 49 by 7 results in 7. This calculated mean gives us a good indication of the 'average' value of the dataset. However, it's important to note that the mean can be heavily influenced by extreme values or outliers in the data, which might not accurately represent the overall dataset.

Finding the Median

The median is the middle number in a sorted list of numbers. To find the median, you arrange the numbers in order from least to greatest and identify the number that sits in the central position. This measure of central tendency is less affected by outliers and skewed data compared to the mean.

In the given exercise, after sorting the numbers (2, 4, 6, 7, 10, 10, 10), we find that 7 is the number right in the middle, with three numbers on either side of it. This makes 7 the median. If there were an even number of data points, the median would be calculated by taking the mean of the two middlemost numbers. Understanding the median is essential because it can provide a better sense of 'typical' for skewed distributions, representing the center of the dataset more accurately in these cases.

Determining the Mode

The mode is perhaps the simplest measure of central tendency to understand and identify. It is the number that appears most frequently in a given set of data. A dataset can have one mode, more than one mode, or no mode at all if no number repeats.

In the exercise example, the sequence of numbers provided is 4, 2, 10, 6, 10, 7, 10. The number 10 appears three times, which is more than any other number, making it the mode of the dataset. Recognizing the mode is especially helpful in understanding the frequency distribution of qualitative data, such as survey responses, and can highlight the most common category or preference in a set of data.