Which measure of central tendency is affected by every score in the distribution

Q: Which measure of central tendency is generally affected by extreme scores?

Arithmetic mean takes into account the value of all items (i.e. very large and very small) in a series. Thus, it is only arithmetic mean which is affected by extreme values in the series.

Q: Which measure of central tendency is best to use when all scores are nominal?

The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency (as it is the only one appropriate to use) when dealing with nominal data.

Q: Which measure of central tendency is the most common value of the variable?

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. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data.

Q: Which measure of central tendency is not affected by extreme scores?

The median is not influenced by extreme values. The median is sensitive only to the value of the middle point or points; it is not sensitive to the values of all other points.

Published on July 30, 2020 by Pritha Bhandari. Revised on November 18, 2022.

Inhaltsverzeichnis Show

Distributions and central tendency
Normal distribution
Skewed distributions
When to use the mode
What can proofreading do for your paper?
Median of an odd-numbered dataset
Median of an even-numbered dataset
Outlier effect on the mean
Population versus sample mean
Frequently asked questions about central tendency
Which measure of central tendency is generally affected by extreme scores?
Which measure of central tendency is best to use when all scores are nominal?
Which measure of central tendency is the most common value of the variable?
Which measure of central tendency is not affected by extreme scores?

Measures of central tendency help you find the middle, or the average, of a dataset. The 3 most common measures of central tendency are the mode, median, and mean.

Mode: the most frequent value.
Median: the middle number in an ordered dataset.
Mean: the sum of all values divided by the total number of values.

In addition to central tendency, the variability and distribution of your dataset is important to understand when performing descriptive statistics.

Distributions and central tendency

A dataset is a distribution of n number of scores or values.

Normal distribution

In a normal distribution, data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The mean, mode and median are exactly the same in a normal distribution.

Example: Normal distributionYou survey a sample in your local community on the number of books they read in the last year.

A histogram of your data shows the frequency of responses for each possible number of books. From looking at the chart, you see that there is a normal distribution.

The mean, median and mode are all equal; the central tendency of this dataset is 8.

Skewed distributions

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other. The direction of this tail tells you the side of the skew

In a positively skewed distribution, there’s a cluster of lower scores and a spread out tail on the right. In a negatively skewed distribution, there’s a cluster of higher scores and a spread out tail on the left.

In this histogram, your distribution is skewed to the right, and the central tendency of your dataset is on the lower end of possible scores.

In a positively skewed distribution, mode < median < mean.

In this histogram, your distribution is skewed to the left, and the central tendency of your dataset is towards the higher end of possible scores.

In a negatively skewed distribution, mean < median < mode.

Mode

The mode is the most frequently occurring value in the dataset. It’s possible to have no mode, one mode, or more than one mode.

To find the mode, sort your dataset numerically or categorically and select the response that occurs most frequently.

Example: Finding the modeIn a survey, you ask 9 participants whether they identify as conservative, moderate, or liberal.

To find the mode, sort your data by category and find which response was chosen most frequently.

To make it easier, you can create a frequency table to count up the values for each category.

Political ideology	Frequency
Conservative	2
Moderate	3
Liberal	4

Mode: Liberal

The mode is easily seen in a bar graph because it is the value with the highest bar.

When to use the mode

The mode is most applicable to data from a nominal level of measurement. Nominal data is classified into mutually exclusive categories, so the mode tells you the most popular category.

For continuous variables or ratio levels of measurement, the mode may not be a helpful measure of central tendency. That’s because there are many more possible values than there are in a nominal or ordinal level of measurement. It’s unlikely for a value to repeat in a ratio level of measurement.

Example: Ratio data with no modeYou collect data on reaction times in a computer task, and your dataset contains values that are all different from each other.

Participant	1	2	3	4	5	6	7	8	9
Reaction time (milliseconds)	267	345	421	324	401	312	382	298	303

In this dataset, there is no mode, because each value occurs only once.

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The median of a dataset is the value that’s exactly in the middle when it is ordered from low to high.

Example: Finding the medianYou measure the reaction times of 7 participants on a computer task and categorize them into 3 groups: slow, medium or fast.

Participant	1	2	3	4	5	6	7
Speed	Medium	Slow	Fast	Fast	Medium	Fast	Slow

To find the median, you first order all values from low to high. Then, you find the value in the middle of the ordered dataset—in this case, the value in the 4th position.

Ordered dataset	Slow	Slow	Medium	Medium	Fast	Fast	Fast

Median: Medium

In larger datasets, it’s easier to use simple formulas to figure out the position of the middle value in the distribution. You use different methods to find the median of a dataset depending on whether the total number of values is even or odd.

Median of an odd-numbered dataset

For an odd-numbered dataset, find the value that lies at the

position, where n is the number of values in the dataset.

ExampleYou measure the reaction times in milliseconds of 5 participants and order the dataset.

Reaction time (milliseconds)	287	298	345	365	380

The middle position is calculated using

, where n = 5.

That means the median is the 3rd value in your ordered dataset.

Median: 345 milliseconds

Median of an even-numbered dataset

For an even-numbered dataset, find the two values in the middle of the dataset: the values at the

and

positions. Then, find their mean.

ExampleYou measure the reaction times of 6 participants and order the dataset.

Reaction time (milliseconds)	287	298	345	357	365	380

The middle positions are calculated using

and

, where n = 6.

That means the middle values are the 3rd value, which is 345, and the 4th value, which is 357.

To get the median, take the mean of the 2 middle values by adding them together and dividing by 2.

Median: 351 milliseconds

Mean

The arithmetic mean of a dataset (which is different from the geometric mean) is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency because all values are used in the calculation.

Example: Finding the mean

Participant	1	2	3	4	5
Reaction time (milliseconds)	287	345	365	298	380

First you add up the sum of all values:

Then you calculate the mean using the formula

There are 5 values in the dataset, so n = 5.

Mean (x̄): 335 milliseconds

Outlier effect on the mean

Outliers can significantly increase or decrease the mean when they are included in the calculation. Since all values are used to calculate the mean, it can be affected by extreme outliers. An outlier is a value that differs significantly from the others in a dataset.

Example: Mean with an outlierIn this dataset, we swap out one value with an extreme outlier.

Participant	1	2	3	4	5
Reaction time (milliseconds)	832	345	365	298	380

Due to the outlier, the mean (

) becomes much higher, even though all the other numbers in the dataset stay the same.

Mean: 444 milliseconds

Population versus sample mean

A dataset contains values from a sample or a population. A population is the entire group that you are interested in researching, while a sample is only a subset of that population.

While data from a sample can help you make estimates about a population, only full population data can give you the complete picture.

In statistics, the notation of a sample mean and a population mean and their formulas are different. But the procedures for calculating the population and sample means are the same.

Sample mean formulaThe sample mean is written as M or x̄ (pronounced x-bar). For calculating the mean of a sample, use this formula:

x̄: sample mean
: sum of all values in the sample dataset
n: number of values in the sample dataset

Population mean formulaThe population mean is written as μ (Greek term mu). For calculating the mean of a population, use this formula:

μ: population mean
: sum of all values in the population dataset
N: number of values in the population dataset

The 3 main measures of central tendency are best used in combination with each other because they have complementary strengths and limitations. But sometimes only 1 or 2 of them are applicable to your dataset, depending on the level of measurement of the variable.

The mode can be used for any level of measurement, but it’s most meaningful for nominal and ordinal levels.
The median can only be used on data that can be ordered – that is, from ordinal, interval and ratio levels of measurement.
The mean can only be used on interval and ratio levels of measurement because it requires equal spacing between adjacent values or scores in the scale.

Levels of measurement

Examples

Measure of central tendency

Nominal

Ethnicity
Political ideology

Mode

Ordinal

Level of anxiety
Income bracket

Mode
Median

Interval and ratio

Reaction time
Test score
Temperature

Mode
Median
Mean

To decide which measures of central tendency to use, you should also consider the distribution of your dataset.

For normally distributed data, all three measures of central tendency will give you the same answer so they can all be used.

In skewed distributions, the median is the best measure because it is unaffected by extreme outliers or non-symmetric distributions of scores. The mean and mode can vary in skewed distributions.

Frequently asked questions about central tendency

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Which measure of central tendency is generally affected by extreme scores?

Arithmetic mean takes into account the value of all items (i.e. very large and very small) in a series. Thus, it is only arithmetic mean which is affected by extreme values in the series.

Which measure of central tendency is best to use when all scores are nominal?

The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency (as it is the only one appropriate to use) when dealing with nominal data.

Which measure of central tendency is the most common value of the variable?

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. Median is the preferred measure of central tendency when: There are a few extreme scores in the distribution of the data.

Which measure of central tendency is not affected by extreme scores?

The median is not influenced by extreme values. The median is sensitive only to the value of the middle point or points; it is not sensitive to the values of all other points.