In everyday language, "average" typically denotes something ordinary, typical, or normal. At times, it can even carry a negative connotation, suggesting mediocrity or even, inadequacy.
In mathematics and statistics, however, the concept of averages holds significant importance, and diverges from its everyday usage. If you are taking statistics for beginners, understanding what averages are, how to use them, and the pros and cons of each type of average is crucial.
In this introduction to statistics, we look at how to work out an average, what is a mode, and the difference between mean,median,mode and range averages.
What Does Mathematics Define as Average?
Hey diddle diddle, the median’s the middle, add and divide for the mean. The mode is the one you see the most and the range is the difference between. Unknown
In mathematics and statistics, the concept of average not only holds utility but also lacks the negative undertones associated with its everyday usage.
Mathematical averages like mean, median, mode and range serve the purpose of condensing multiple values into a single representation, offering insight into the meaning of various sets of data.
In addition, averages prove beneficial for summarising multiple datasets and facilitating comparisons. Especially when faced with an abundance of numerical information, averages serve as invaluable tools for simplification and comprehension.
The Four Types of Averages: Mean, Median, Mode and Range
Knowing how to work out an average means understanding the types. In mathematics and statistics, four primary types of averages exist: the mean, median, mode, and range. Each of these averages yields distinct results for the same dataset, and this variability is beneficial.
The diverse range of averages enables their application across various types of data and for different analytical purposes. However, due to its unique nature compared to mean medium mode, we won't delve into the range in this article.
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What is the Mean?
The mean, the most commonly known method of how to work out an average, is typically the go-to method for calculating an "average" in most contexts.
It's sometimes denoted by the Greek letter "μ" and is computed by summing up all the numbers in the dataset and dividing by the total count of instances.
Using the formula:
Mean = Sum of values⁄Number of values
For instance, let's consider the highest temperatures recorded in a certain city for the week (at the time of writing): 32, 24, 18, 19, 17, 17, and 18.
- First, we add these numbers together, resulting in a sum of 145, which represents the "Sum of values".
- Next, we determine the total count of values, which in this case is 7, as we have data for each day of the week. This represents our "Number of values". • Dividing 145 by 7 (145/7) gives us 20.7143. However, since our original temperatures were rounded to the closest degree, we round this figure to maintain consistency with our data format.
While rounding is optional, in this case, the average highest temperature for the week is "21 degrees".
Upon reviewing the numbers, one may intuitively feel that this average aligns well with the dataset. If any of these temperatures were the highest recorded for any given day that week, it wouldn't seem unusual.
The Advantages of the Mean
There are several advantages to using the mean. Firstly, it incorporates and considers every value within the dataset, ensuring that each value contributes to the final outcome.
Furthermore, the mean is straightforward and user-friendly, especially when dealing with smaller datasets. Even with larger datasets, it remains manageable, although using a spreadsheet can streamline the process when dealing with extensive data.
The insights derived from the mean can be further analysed, making it a preferred choice for many mathematicians. The mean offers a more precise measurement as it can fall between the numbers or values within your dataset. In contrast, with the median and mode, as we'll discuss shortly, the results are limited to being one of the values or midway between two values.
Additionally, the mean acts as a smoothing mechanism for datasets containing outliers. In our example, despite having one exceptionally warm day (32 degrees), the mean temperature was still considerably cooler, effectively averaging out the extremes.
The Disadvantages of the Mean
However, the mean does have its drawbacks. While it helped mitigate the impact of outliers, the data still reflects their influence. If we were to exclude the particularly hot day, the average temperature would decrease to approximately 18.83 or 19 degrees when rounded.
Furthermore, the mean is primarily applicable to numerical data. Attempting to represent non-numerical data with numbers may render the mean less meaningful or even irrelevant.
Additionally, the scale used can affect the mean. For instance, if we consider the same highest temperatures in Fahrenheit (90, 76, 64, 66, 63, 63, and 65), the average becomes 72°F or approximately 22°C. Thus, the choice of scale influences the resulting mean.
What is a Mode?
The mode is our second "average". It represents the most frequent occurrence in a set of data. Basically, this is the number that appears most in the data or the value with the highest frequency.
Let's take the weather example again. We had 32, 24, 18, 19, 17, 17, and 18. You can quite easily calculate the mode without needing to do anything complicated because we only have 7 numbers to consider.
In this example, the mode is "17" as it appears twice in our set whereas every other number only appears once each time.
By taking the mode, our average is "17", which is 3 degrees off our mean.
The mode can give you multiple results and not necessarily numbers that are close to each other.
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The Advantages of the Mode
So, with, what is a mode, behind us, there are a few advantages of the mode. For one, it's probably one of the easiest ones to calculate as you only need to count the various instances and while this can be time-consuming with big sets of data, with a few numbers, you can do it quite easily without any maths or a calculator.
It's also useful for certain sets of data where the other averages' disadvantages may render them fairly meaningless or useless.
The mode is less affected by outliers, too, as in our example here, it wouldn't make any difference if the 32-degree day was 35, 40, or even 50. The mode would still be 17.
You can still find an average from a set that isn't numerical with the mode, making it useful for finding popular choices or votes from a list. It's also particularly useful in statistics for non-numerical data and finding a tendency in sets of data whose distribution is clustered around several different points.
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The Disadvantages of the Mode
You can probably guess some of the disadvantages of the mode. In our data, it only took 2 instances for "17" to become the mode. For sets of data with lots of different numbers, it would only take a small number of instances for that particular number to become the mode.
Imagine we took the highest temperatures across the year and almost every day was a different temperature, except for a few days when the high was 40 degrees. This being the average wouldn't be particularly helpful in working out what to bring when visiting, would it?
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What is the Median?
The median is the third type of average we can use.
It is used to determine the "middle" value in a set of data.
Calculating the median is relatively straightforward. Simply arrange all the numbers numerically, identify the value positioned in the middle, and that becomes your median.
In our example, with 7 values (32, 24, 18, 19, 17, 17, and 18), we arrange them in ascending order: 17, 17, 18, 18, 19, 24, 32. Since there are 7 values, the fourth value (18) is in the middle, giving us a median of 18.
In datasets with an even number of values, you take the two values sharing the middle position and calculate their mean, which represents the halfway point between the two values.
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The Advantages of the Median
The median offers the advantage of significantly mitigating the impact of outliers. Even if the two hottest days were twice as hot, our median would still be 18. When the data distribution is uneven, the median should be used to minimise the influence of extremely high or low values. The median is commonly used for calculating averages in scenarios such as average salaries, house prices, and other datasets susceptible to skewing by exceptionally high values. For instance, if the mean were used to calculate the average salary for South Africans, the presence of a handful of billionaires and millionaires—whose salaries are orders of magnitude greater than the typical South African—would distort the perception of the actual earnings.
The Disadvantages of the Median
The median disregards some data, making it unsuitable for reflecting extreme values in the average. Its precision is limited since it uses only one data point, neglecting the degrees between values. In uneven distributions, this can lead to a significant deviation from the average. While manageable for small datasets like ours with 7 values, larger datasets may pose challenges.
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