Let's talk about measures of central tendency! If you're a data lover (and even more if you are not), you know that sometimes a bunch of numbers can be overwhelming. That's where measures of central tendency come in - they help us make sense of our data by giving us a summary of the data group. Whether you're calculating the mean, median, or mode, these measures provide a helpful snapshot that can guide decision-making and analysis. But beware - outliers and skewed distributions can throw off our measures, so it's important to consider the bigger picture.
So, grab your calculator and let's dive into the world of measures of central tendency! Keep reading to learn more.
What is central tendency?
A measure of central tendency attempts to describe a dataset through a singular value. This singular value is meant to represent the center point or typical value in a dataset.
There are three measures of central tendency we need to know about:
- the mean (also referred to as the average),
- the median and
- the mode
Mean as a Measure of Central Tendency
The mean represents the average value of a set of data.
The process to find the mean is to sum all the values of the data set, and then divide that by the number of data points. The mean is usually represented by the Greek letter \(\mu\), pronounced /mi/:
\[\frac{value_1 + value_2 + value_3 +...+ value_n}{n}\]
This can also be expressed with the following equation:
\[\frac{\sum^{n}_{i=1} x_i}{n}\]
This means that all values (represented with x) starting at value number 1 (i = 1) will be added until we reach value n, and then this number will be divided by n (the total number of values).
i = 1 does not mean that only values starting at 1 are counted, but rather that the sum starts at the first value. The first value, depending on how the list of numbers is ordered, could be 1 or any other value.
The mean is one of the most commonly used measures of central tendency in data analysis and is often used to summarize data sets. However, the mean can be influenced by extreme values in the data set, known as outliers, which can skew the results. In such cases, other measures of central tendency, such as the median or mode, may be more appropriate.
Find the mean value of rainfall for the days listed below
Day | 1 | 2 | 3 | | 5 | | | | 9 | 10 |
Rainfall (mm) | 10 | | 0 | 5 | | 2 | 29 | 1 | | |
Solution
The mean is given by the sum of all the values divided by the number of values.
\[\frac{\sum^{n}_{i=1} x_i}{n} = \frac {\sum^{10}_{i=1} value_i}{10} = (\frac{10 + 12 + 0 + 5 + 17 + 2 + 29 + 1 + 4 + 14)}{1}0 = \frac{94}{10} = 9.4 mm\]
Median as a Measure of Central Tendency
The median is a measure of central tendency that represents the value separating the higher half from the lower half of a data set. If the data set has an even number of values, the median is the average of the two middle values.
When we have a set of data that is able to be ordered in some way, we can find the median. The process to find the median is as follows:
- Order the data, from smallest to largest.
- If the number of data points is odd, the middle number is the median, meaning we take the \((\frac{n+1}{2})^{th}\) value.
- If the number of data points is even, then we take the mean value of the middle two values. This means we take mean of the \((\frac{n}{2})^{th}\) and \((\frac{n+2}{2})^{th}\) value.
- Find the median of the following data.
12, 3, 4, 7, 19, 13, 4, 8, 81
Solution
The first thing we need to do is order the data from smallest to largest, and this results in
3, 4, 4, 7, 8, 12, 13, 19, 81
As this has an odd number of data points, the median is the middle number of the ordered dataset, giving a median of 8.
Given below are the heights of 30 children in a class (height given in cm). Find the median height.
168, 172, 151, 145, 181, 162, 174, 159, 149, 180, 164, 171, 150, 143, 189, 167, 176, 156, 144, 186, 166, 177, 153, 140, 184, 163, 178, 158, 149, 187.
Solution
First of all, we must order the data from smallest to largest. We get:
140, 143, 144, 145, 149, 149, 150, 151, 153, 156, 158, 159, 162, 163, 164, 166, 167, 168, 171, 172, 174, 176, 177, 178, 180, 181, 184, 186, 187, 189.
As thirty is even, to find the median we find the mean of the fifteenth and sixteenth values. The fifteenth value is 164, and the sixteenth value is 166. The mean of these values is \(\frac{164+166}{2} = 165\), meaning the median value is 165.
Mode as a Measure of Central Tendency
The mode is a measure of central tendency that represents the most frequently occurring value in a data set. It is often used in combination with the mean and median to give a complete picture of a data set. A data set may have multiple modes if there are two or more values that occur with equal frequency. The mode is a useful statistic for identifying the most common value in a data set, and can be particularly informative in situations where the data is discrete (i.e. consisting of whole numbers or categories) rather than continuous (i.e. consisting of a range of values).
The mode of a set of data is the most common value in the dataset. If there are two or more values which are most common, both of these values are the mode.
Find the mode of the following data set.
1, 2, 3, 4, 4, 5, 6, 6, 6, 6, 7
Solution
The mode here would be 6, as this appears four times, which makes it the most common value.
Find the mode of the following numbers.
1, 2, 2, 3, 3, 3, 5, 7, 7, 7, 9, 11, 134
Solution
Both 3 and 7 appear three times, making them both the most common value, meaning the mode is 3 and 7.
Choosing suitable measures of central tendency
Each measure of central tendency has its own advantages and disadvantages.
For the mean, the advantages are that it uses all of the data, and is, therefore, representative of all the data. However, there are disadvantages to using the mean. It is disproportionately influenced by extreme values, which can throw the mean. The mean also cannot be used if our data isn't numerical, and takes the most computation out of all our measures of central tendency.
For the mode, the advantages are that we can find the mode of a set of data, be it numeric or otherwise. There is also limited computation, as we only need to tally the data, meaning if our data comes pre-tallied then this aids the mode. However, a downside is that the mode doesn't necessarily exist. In addition, we can have multiple modes, which doesn't help us describe a lot about the data set. As well as this, the mode doesn't take into account the full data set.
Our final measure of central tendency is the median. The advantages are that the median isn't affected by any outliers or extreme values, and we have very little calculation to do. On the flip side, it does require us to order the set of data, which for large sets of data, is lengthy and time-consuming. It also doesn't take into account the full set of data, which means this could bring in weak results.
Measures of Central Tendency - Key takeaways
To find the mean we add up all the values in the data set and divide by the number of data points.
The formula for the mean is \[\frac{\sum^{n}_{i=1} x_i\}{n}\]
The mode is the most common value in a data set.
The median is the central value of the data set.
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