Which measure of central tendency is affected most by the scores of these two students
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The concept of central tendency is first introduced to Singapore students taking mathematics in secondary school. For many, this theorem is of great interest as it is apparent how it influences the way we see the world. Equally of interest is that it is widely used in a multitude of tertiary education courses and real-life careers. What is central tendency, what preceded it and how is it used in everyday life? In this article, we will be exploring all 3 of these questions. What is Central TendencyThe “measures of central tendency” is a mathematical concept under the branch of mathematics known as statistics. In the simplest of terms, it attempts to find a single value that best represents an entire distribution of scores. In doing so, concepts of “average value”, “mid-point” and “most common value” are used in an effort to represent the distribution. Students will find these descriptions familiar as they are descriptive forms of the measures known as “mean”, “median” and “mode”. The Importance of Central TendencyAt its core, mathematics and statistics is a language. When we have a range of results, we require an efficient manner to communicate it to others. Rather than listing every single score in the population, a “summarised version” would be preferred. This is why having a single value to present the distribution is required and easier to comprehend. These same measures can then be used to make estimations and efficiently work on a population. Early History of Average ValuesHistorical stories and recounts have featured efforts by people of that time to perform estimation. In doing so, they relied on intuitively finding a measure of centre. Here are 2 examples that have been recorded. The Athenians’ Wall ConquestIn Greek history, records of the Athenians have shown them using a form of estimation to climb over their enemies’ fortress walls. To get into the forts of their enemies, Athenians would construct ladders with sufficient height to reach the top of their walls. However, to find the right height, they had their men count the number of bricks in a column of a wall. Naturally, given the conditions, they returned with many different answers. While naturally some was wrong, it was accepted that the majority had to be right. In this case, we can see an implicit use of “mode”. The Athenians assumed that the most frequent value had to be correct. This was true even if none of the values represented more than half of the men. The Greeks’ Estimation of SailorsReferring again to an account from the Greeks, who this time were attempting to count the number of sailors in their fleet. This was done by taking the known largest and smallest ship sailor numbers before finding the average and multiplying it by the number of ships. The Greeks assumed that the size of ship crews would be fairly evenly spread out between the smallest and largest. Furthermore, it did not matter that the average was not a whole number. In this case, we can see an implicit use of the “arithmetic mean”. Rather than any observable value to represent the distribution, the Greeks used a purely derived number as the average. Before the Measures of Central Tendency were FormalisedGiven the early usage of similar methods to that of Central Tendency, you can be forgiven for wondering what is its true value. To help illustrate this, let us have a look at statements provided by many students when asked: “what is the average?”
Yet again, from these responses, we can see that we all have an inkling of what the representation of a distribution should be. We can also acknowledge that there is no one true way of determining the ideal representation. The Value in Central TendencySince there are multiple ways to determine the average and make inferences from it, how then do we choose which to use? This is where Central Tendency is needed. By recognising the patterns of distributions as well as the characteristics of each measure, we can select the most appropriate measure to be used. As we introduce the different measures of Central Tendency, it is important to bear in mind the objective. This is to find a measure that is representative of the average of the distribution. ModeThe mode as illustrated by the Greeks in our earlier example, is simply the most frequently occurring score. The advantages of using mode include:
The disadvantages of using mode include:
MedianDescribed as the mid-point, the meridian divides a distribution in half. This means that half of the scores will be less than the median while the other 50% will be above it. Based on this concept, we can see how a distribution with an odd number of data points easily fits in. However, things are not as clear for distributions with an even number of data points. As such, here is how the median is found in both cases:
The advantages of using median include:
The disadvantages of using median include:
Arithmetic MeanIn mathematics, there are several forms of the mean such as the arithmetic, geometric and harmonic mean. We will be focusing on the arithmetic mean here which is often described as the “average”. The arithmetic mean is a highly popular statistic due to its simplicity and versatility. In contrast to the median, the mean uses all values in the distribution. All data points are added up before dividing this sum by the number of data points. In doing so, an arithmetic average value is found whereby it is not a value that needs to be equal to any 1 data point. The advantages of using mean include:
The disadvantages of using mean include:
Conclusion on the meanAll in all, the arithmetic mean is considered to be the most reliable measure of Central Tendency. Save for the situations listed above where median or more is more advantageous, you will often find that the mean is the most reliable representation of a distribution. Einstein’s Takeaway:Understanding the characteristics of a distribution and choosing the most appropriate measure of Central Tendency is critical for understanding and communicating about a data set. This is why so much emphasis is placed on this topic. Furthermore, at further levels of academia or careers, the measures of central tendency are seldom used alone. Oftentimes, they used in tandem with the variance or standard deviation of a sample in order to further understand it or to plan a distribution. Given its continual use and heavy usage in daily life, students should be familiar with the measures of Central Tendency as well as their applications. This is where sec 3 E math tuition and sec 3 A math tuition are valuable. Having the extra attention by qualified tutors will allow you or your child to fully grasp the measures of Central Tendency. Contact Einstein Education Hub today to find out how we can help! Which measure of central tendency is affected by every score in the distribution?However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean.
Which measure of central tendency is most influenced by extreme scores?The measure of central tendency which is most strongly influenced by extreme values in the 'tail' of the distribution is: the mean.
Which measure of central tendency is most likely to be affected by one or two extreme scores in a distribution?The mean is the measure of central tendency most likely to be affected by an extreme value. Mean is the only measure of central tendency which depends on all the values as it is derived from the sum of the values divided by the number of observations.
Which measure of central tendency would change the most if an additional test score of 2 was included in the distribution?Which measure of central tendency would change the most if an additional test score of 2 was included in the distribution? The median would change the most.
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