4.4.1 Mean, median & mode
Average:
In general, the term “average” refers to a measure that represents the typical or central value of a set of numbers or data points. There are different types of averages, but the most common ones are mean, median, and mode.
Mean:
The mean is the most commonly used average, and it is calculated by adding all the values in a dataset and dividing by the number of values.
It is also referred to as the arithmetic mean.
The formula for calculating the mean is:
\( Mean = \frac{sum \ of \ all \ values}{number \ of \ values} \)
For example, if you have a dataset of numbers \( {2, 4, 6, 8, 10} \), the mean would be:
\( Mean = \frac{2+4+6+8+10}{5}=6 \)
So, the mean of this dataset is \( 6 \).
The mean is useful in summarizing data because it provides a single value that represents the typical value of the dataset.
Median:
The median is the middle value of a dataset, i.e., the value that separates the higher half from the lower half of the data.
To find the median, you need to order the values in the dataset from lowest to highest and then find the middle value.
If there is an odd number of values in the dataset, the median is the middle value.
If there is an even number of values, the median is the average of the two middle values.
For example, if you have a dataset of numbers \( {2, 4, 6, 8, 10} \)
Arrange the values in order: \( {2, 4, 6, 8, 10} \)
Since there are \( 5 \) values, the median is the middle value, which is \( 6 \). So, the median of this dataset is \( 6 \). The median is useful when the data contains extreme values that can distort the mean, making it less representative of the typical value in the dataset.
Mode:
The mode is the value that appears most frequently in a dataset.
Mode is a type of average that represents the value that occurs most frequently in a dataset.
It is a measure of central tendency that is useful when you want to identify the most common or popular value in a dataset.
For example, if you have a dataset of numbers \( {2, 4, 6, 8, 6, 10, 6} \), the mode would be \( 6 \) because it occurs more frequently than any other value in the dataset.
It is possible for a dataset to have more than one mode if multiple values occur with the same highest frequency. In this case, the dataset is said to be bimodal, trimodal, etc.
The mode is useful in summarizing data because it provides information about the most commonly occurring value, which can be helpful in making decisions or drawing conclusions about the data.
Working with means:
Here are some ways to work with means:
Calculate the mean:
To calculate the mean of a dataset, you need to add up all the values in the dataset and divide by the number of values. This will give you the arithmetic mean, which is the typical value of the dataset.
Compare means:
You can compare the means of different datasets to identify similarities or differences between them. For example, you can compare the mean income of different countries to see which countries have higher or lower incomes.
Identify outliers:
Outliers are values in a dataset that are significantly higher or lower than the other values. Outliers can affect the mean and make it less representative of the typical value in the dataset. By examining the mean and the range of values in the dataset, you can identify potential outliers and decide whether to remove them or investigate further.
Evaluate changes over time:
By calculating the mean of a dataset at different points in time, you can identify trends and evaluate changes over time. For example, you can calculate the mean temperature in a city over a period of several years to see if there is a trend of increasing or decreasing temperatures.
Overall, working with means involves using the arithmetic mean to summarize data, identify trends, and make comparisons between different datasets.
Worked example:
The test scores of \( 14 \) students are shown below.
\( 21 \) \( 21 \) \( 23 \) \( 26 \) \( 25 \) \( 21 \) \( 22 \) \( 20 \) \( 21 \) \( 23 \) \( 23 \) \( 27 \) \( 24 \) \( 21 \)
Find the mode, median and mean of the test score.
The mode is the number which occurs the most frequently.
Here the mode is \( 21 \) as it occurs \( 5 \) times more than any other number.
\( 21 \)
To find median, put the numbers in order
\( 20 \) \( 21 \) \( 21 \) \( 21 \) \( 21 \) \( 21 \) \( 22 \) \( 23 \) \( 23 \) \( 23 \) \( 24 \) \( 25 \) \( 26 \) \( 27 \)
As there are \( 14 \) pieces of data, the middle of the data set will be halfway between \( 7 \)th and \( 8 \)th pieces of data.
Now add \( 7 \)th and \( 8 \)th term and divide the sum by \( 2 \) to get the median.
\( Median=\frac{7^{th} \ term+8^{th} \ term}{2} \)
\( 20 \) \( 21 \) \( 21 \) \( 21 \) \( 21 \) \( 21 \) \( 22 \) \( 23 \) \( 23 \) \( 23 \) \( 24 \) \( 25 \) \( 26 \) \( 27 \)
\( Median=\frac{22+23}{2}=22.5 \)
To find mean, use the formula
\( Mean = \frac{sum \ of \ all \ values}{number \ of \ values} \)
\( Mean = \frac{20+21+21+21+21+21+22+23+23+23+24+25+26+27}{14} \)
\( Mean=\frac{318}{14} \)
\( Mean=22.7 \)
\( Mode=21 \)
\( Median=22.5 \)
\( Mean=22.7 \)