Which measure of central tendency is the value that appears most often in a set of data Brainly?

Learning Outcomes

  • Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.

By now, everyone should know how to calculate mean, median and mode. They each give us a measure of Central Tendency (i.e. where the center of our data falls), but often give different answers. So how do we know when to use each? Here are some general rules:

  1.  Mean is the most frequently used measure of central tendency and generally considered the best measure of it. However, there are some situations where either median or mode are preferred.
  2. Median is the preferred measure of central tendency when:
    1.  There are a few extreme scores in the distribution of the data. (NOTE: Remember that a single outlier can have a great effect on the mean). b.
    2. There are some missing or undetermined values in your data. c.
    3. There is an open ended distribution (For example, if you have a data field which measures number of children and your options are [latex]0[/latex], [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex] or “[latex]6[/latex] or more,” than the “[latex]6[/latex] or more field” is open ended and makes calculating the mean impossible, since we do not know exact values for this field).
    4. You have data measured on an ordinal scale.
  3. Mode is the preferred measure when data are measured in a nominal ( and even sometimes ordinal) scale.

Video transcript

Find the mean, median, and mode of the following sets of numbers. And they give us the numbers right over here. So if someone just says the mean, they're really referring to what we typically, in everyday language, call the average. Sometimes it's called the arithmetic mean because you'll learn that there's other ways of actually calculating a mean. But it's really you just sum up all the numbers and you divide by the numbers there are. And so it's one way of measuring the central tendency. The average, I guess, we could say. So this is our mean. We want to average 23 plus 29-- or we're going to sum 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25, and then divide that by the number of numbers. So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. So you want to divide that by 8. So let's figure out what that actually is. Actually, I'll just get the calculator out for this part. I could do it by hand, but we'll save some time over here. So we have 23 plus 29 plus 20 plus 32 plus 23 plus 21 plus 33 plus 25. So the sum of all the numbers is 206. And then we want to divide 206 by 8. So if I say 206 divided by 8 gets us 25.75. So the mean is equal to 25.75. So this is one way to kind of measure the center, the central tendency. Another way is with the median. And this is to pick out the middle number, the median. And to figure out the median, what we want to do is order these numbers from least to greatest. So it looks like the smallest number here is 20. Then, the next one is 21. There's no 22 here. Let's see, there's two 23's. 23 and a 23. So 23 and a 23. And no 24's. There's a 25. 25. There's no 26, 27, 28. There is a 29. 29. Then you have your 32. 32. And then you have your 33. 33. So what's the middle number now that we've ordered it? So we have 1, 2, 3, 4, 5, 6, 7, 8 numbers. We already knew that. And so there's actually going to be two middles. If you have an even number, there's actually two numbers that qualify for close to the middle. And to actually get the median, we're going to average them. So 23 will be one of them. That, by itself, can't be the median because there's three less than it and there's four greater than it. And 25 by itself can't be the median because there's three larger than it and four less than it. So what we do is we take the mean of these two numbers and we pick that as the median. So if you take 23 plus 25 divided by 2, that's 48 over 2, which is equal to 24. So even though 24 isn't one of these numbers, the median is 24. So this is the middle number. So once again, this is one way of thinking about central tendency. If you wanted a number that could somehow represent the middle. And I want to be clear, there's no one way of doing it. This is one way of measuring the middle. Let me put that in quotes. The middle. If you had to represent this data with one number. And this is another way of representing the middle. Then finally, we can think about the mode. And the mode is just the number that shows up the most in this data set. And all of these numbers show up once, except we have the 23, it shows up twice. And since because 23 shows up the most, it shows up twice. Every other number shows up once, 23 is our mode.

Recommended: First read Measures of Shape


What are the measures of central tendency?

A measure of central tendency (also referred to as measures of centre or central location) is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or centre of its distribution.


There are three main measures of central tendency: the mode, the median and the mean. Each of these measures describes a different indication of the typical or central value in the distribution.


What is the mode?

The mode is the most commonly occurringvalue in a distribution.

Consider this dataset showing the retirement age of 11 people, in whole years:54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60This table shows a simple frequency distribution of the retirement age data.

Age

Frequency

54

3

55

1

56

1

57

2

58

2

60

2


The most commonly occurring value is 54, therefore the mode of this distribution is 54 years. Advantage of the mode:The mode has an advantage over the median and the mean as it can be found for both numerical and categorical (non-numerical) data. Limitations of the mode:The are some limitations to using the mode. In some distributions, the mode may not reflect the centre of the distribution very well. When the distribution of retirement age is ordered from lowest to highest value, it is easy to see that the centre of the distribution is 57 years, but the mode is lower, at 54 years. 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60It is also possible for there to be more than one mode for the same distribution of data, (bi-modal, or multi-modal). The presence of more than one mode can limit the ability of the mode in describing the centre or typical value of the distribution because a single value to describe the centre cannot be identified.In some cases, particularly where the data are continuous, the distribution may have no mode at all (i.e. if all values are different).In cases such as these, it may be better to consider using the median or mean, or group the data in to appropriate intervals, and find the modal class.

What is the median?

The median is the middlevalue in distribution when the values are arranged in ascending or descending order.

The median divides the distribution in half (there are 50% of observations on either side of the median value). In a distribution with an odd number of observations, the median value is the middle value. Looking at the retirement age distribution (which has 11 observations), the median is the middle value, which is 57 years: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 When the distribution has an even number of observations, the median value is the mean of the two middle values. In the following distribution, the two middle values are 56 and 57, therefore the median equals 56.5 years: 52, 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60Advantage of the median:The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical. Limitation of the median:The median cannot be identified for categorical nominal data, as it cannot be logically ordered.

What is the mean?

The mean is the sum of the value of each observation in a dataset divided by the number of observations. This is also known as the arithmetic average.

Looking at the retirement age distribution again: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60The mean is calculated by adding together all the values (54+54+54+55+56+57+57+58+58+60+60 = 623) and dividing by the number of observations (11) which equals 56.6 years.Advantage of the mean:The mean can be used for both continuous and discrete numeric data.Limitations of the mean:The mean cannot be calculated for categorical data, as the values cannot be summed.As the mean includes every value in the distribution the mean is influenced by outliers and skewed distributions.What else do I need to know about the mean?The population mean is indicated by the Greek symbol (pronounced ‘mu’). When the mean is calculated on a distribution from a sample it is indicated by the symbol (pronounced X-bar).

How does the shape of a distribution influence the Measures of Central Tendency?

Symmetrical distributions:When a distribution is symmetrical, the mode, median and mean are all in the middle of the distribution. The following graph shows a larger retirement age dataset with a distribution which is symmetrical. The mode, median and mean all equal 58 years.

Skewed distributions:When a distribution is skewed the mode remains the most commonly occurring value, the median remains the middle value in the distribution, but the mean is generally ‘pulled’ in the direction of the tails. In a skewed distribution, the median is often a preferred measure of central tendency, as the mean is not usually in the middle of the distribution. A distribution is said to be positively or right skewed when the tail on the right side of the distribution is longer than the left side. In a positively skewed distribution it is common for the mean to be ‘pulled’ toward the right tail of the distribution. Although there are exceptions to this rule, generally, most of the values, including the median value, tend to be less than the mean value. The following graph shows a larger retirement age data set with a distribution which is right skewed. The data has been grouped into classes, as the variable being measured (retirement age) is continuous. The mode is 54 years, the modal class is 54-56 years, the median is 56 years and the mean is 57.2 years.

A distribution is said to be negatively or left skewed when the tail on the left side of the distribution is longer than the right side. In a negatively skewed distribution, it is common for the mean to be ‘pulled’ toward the left tail of the distribution. Although there are exceptions to this rule, generally, most of the values, including the median value, tend to be greater than the mean value. The following graph shows a larger retirement age dataset with a distribution which left skewed. The mode is 65 years, the modal class is 63-65 years, the median is 63 years and the mean is 61.8 years.


How do outliers influence the measures of central tendency?

Outliers are extreme, or atypical data value(s) that are notably different from the rest of the data.

It is important to detect outliers within a distribution, because they can alter the results of the data analysis. The mean is more sensitive to the existence of outliers than the median or mode. Consider the initial retirement age dataset again, with one difference; the last observation of 60 years has been replaced with a retirement age of 81 years. This value is much higher than the other values, and could be considered an outlier. However, it has not changed the middle of the distribution, and therefore the median value is still 57 years. 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 81As the all values are included in the calculation of the mean, the outlier will influence the mean value. (54+54+54+55+56+57+57+58+58+60+81 = 644), divided by 11 = 58.5 yearsIn this distribution the outlier value has increased the mean value. Despite the existence of outliers in a distribution, the mean can still be an appropriate measure of central tendency, especially if the rest of the data is normally distributed. If the outlier is confirmed as a valid extreme value, it should not be removed from the dataset. Several common regression techniques can help reduce the influence of outliers on the mean value.Return to Statistical Language Homepage


Further information:

External links:

easycalculation.com - Mean, Median, Mode Calculator
calculatorsoup.com - Descriptive Statistics calculator
calculatorsoup.com - Mean Median Mode calculator

Which measure of central tendency is the value that appears most often in a set of data?

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. For data from skewed distributions, the median is better than the mean because it isn't influenced by extremely large values.

Which measure of central tendency is the value that appears most often in a set of data a median B mode C mean?

There are three main measures of central tendency: the mode, the median and the mean. Each of these measures describes a different indication of the typical or central value in the distribution. What is the mode? The mode is the most commonly occurring value in a distribution.

Which measure of central tendency is the middle value in a set of ordered data Brainly?

The median is the middle number in an ordered data set. The mean is the sum of all values divided by the total number of values.

What is the most commonly used measure of central tendency?

Mean is the most commonly used measure of central tendency. There are different types of mean, viz. arithmetic mean, weighted mean, geometric mean (GM) and harmonic mean (HM).