Numerical Summaries of Data: Learn It 1

  • Find the average, middle value, and most common value in a set of data
  • Calculate how spread out the data is using the range and standard deviation
  • Identify the parts of a five-number summary for a set of data and create a box plot

Mean, Median, and Mode

When we talk about the “center” of a data set in statistics, we are often referring to measures of central tendency, which summarize a key aspect of the distribution of the data. The mean and median are two such measures, each providing a different perspective on the data.  Understanding which measure to use gives us insight into the true nature of the data’s central tendency.

Mean

In analyzing quantitative data, the measure of center will be one key component. The mean, calculated as the average of all values, can be influenced by outliers and skewed distributions. 

mean

The mean of a set of [latex]n[/latex] numbers is the arithmetic average of the numbers.

 

[latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]

How To: Calculate the Mean of a Set of Numbers

  1. Write the formula for the mean:

    [latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]

  2. Find the sum of all the values in the set. Write the sum in the numerator.
  3. Count the number, [latex]n[/latex], of values in the set. Write this number in the denominator.
  4. Simplify the fraction.
  5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

Median

The median, the middle value when all observations are ordered, is more robust to outliers and provides a better representation of the “typical” value in a skewed dataset.

median

The median of a set of data values is the middle value.

 

  • Half the data values are less than or equal to the median.
  • Half the data values are greater than or equal to the median.

How To: Find the Median of a Set of Numbers

  1. List the numbers from smallest to largest.
  2. Count how many numbers are in the set. Call this [latex]n[/latex].
  3. Is [latex]n[/latex] odd or even?
    • If [latex]n[/latex] is an odd number, the median is the middle value.
    • If [latex]n[/latex] is an even number, the median is the mean of the two middle values.
Let’s consider this small set of data values:

[latex]3.3\qquad 0.8\qquad 5.8\qquad 10.0\qquad 3.6\qquad 8.7\qquad 0[/latex]

a) Calculate the mean of the data set.

Mean [latex]= 4.6[/latex]

 

b) Calculate the median of the data set.

Median [latex]= 3.6[/latex]

Mode

In addition to mean and median, the mode is another measure of central tendency that identifies the most common or frequent value in a data set. It can be particularly useful in understanding the distribution of categorical data, where numerical averages are not applicable.

mode

The mode of a set of numbers is the number with the highest frequency.

How To: Find the Mode of a Set of Numbers

  1. List the data values in numerical order.
  2. Count the number of times each value appears.
  3. The mode is the value with the highest frequency.

Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

Statistics exam scores for 2020 students are as follows:

[latex]50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93[/latex]

Find the mode.

The data lists the heights (in inches) of students in a statistics class. Identify the mode.

[latex]56[/latex] [latex]61[/latex] [latex]63[/latex] [latex]64[/latex] [latex]65[/latex] [latex]66[/latex] [latex]67[/latex] [latex]67[/latex]
[latex]60[/latex] [latex]62[/latex] [latex]63[/latex] [latex]64[/latex] [latex]65[/latex] [latex]66[/latex] [latex]67[/latex] [latex]70[/latex]
[latex]60[/latex] [latex]63[/latex] [latex]63[/latex] [latex]64[/latex] [latex]66[/latex] [latex]66[/latex] [latex]67[/latex] [latex]74[/latex]
[latex]61[/latex] [latex]63[/latex] [latex]64[/latex] [latex]65[/latex] [latex]66[/latex] [latex]67[/latex] [latex]67[/latex]  

Mean, Median, and Mode

 

You can view the transcript for “Math Antics – Mean, Median and Mode” here (opens in new window).

You can view the transcript for “Finding mean, median, and mode | Descriptive statistics | Probability and Statistics | Khan Academy” here (opens in new window).

You can view the transcript for “Mean, Median, Mode, and Range | Math with Mr. J” here (opens in new window).