Measure of Central Tendency

Introduction to Measure of Central Tendency

The Measure of central tendency is a statistic that summarizes the entire quantitative or qualitative set of data in a single value (a representative value of the data set) tending to concentrate somewhere in the center of the data. The tendency of the observations to cluster in the central part of the data is called the central tendency and the summary values as measures of central tendency, also known as the measure of location or position, are also known as averages.

Note that

• The Measure of central tendency should be within the data set’s range.
• It should remain unchanged by rearranging the observations in a different order.

Criteria of Satisfactory Measures of Location or Averages

There are several types of averages available to measure the representative value of a set of data or distribution. So, an average should satisfy or possess all or most of the following conditions.

• It should be well-defined, i.e., rigorously defined. There should be no confusion in its definition. The sum of values divided by their total number is the well-defined definition of Arithmetic Mean.
• It should be based on all the observations made.
• It should be simple to understand and easy to interpret.
• It can be calculated quickly and easily.
• It should be amenable/manageable to mathematical treatment.
• It should be relatively stable in repeating sampling experiments.
• It should not be unduly influenced by abnormally large or small observations (i.e., extreme observations)

The mean, median, and mode are all valid measures of central tendencies, but under different conditions, some measures of central tendencies become more appropriate to use than others. There are several different kinds of calculations for central tendency, where the kind of calculation depends on the type of the data, i.e. level of measurement on which data is measured.

Measures of Central Tendencies

The following are the measures of central tendencies for univariate or multivariate data.

• The arithmetic mean: The sum of all measurements divided by the number of observations in the data set
• Median:  The middlemost value for sorted data. The median separates the higher half from the lower half of the data set, i.e., partitioning the data set into parts.
• Mode: The most frequent or repeated value in the data set.
• Geometric mean: The nth root of the product of the data values.
• Harmonic mean: The reciprocal of the arithmetic mean of the reciprocals of the data values
• Weighted mean: An arithmetic mean incorporating the weights to elements of certain data.
• Distance-weighted estimator: The measure uses weighting coefficients for $x_i$ that are computed as the inverse mean distance between $x_i$ and the other data points.
• Truncated mean: The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
• Midrange: The arithmetic mean of the maximum and minimum values of a data set.
• Midhinge: The arithmetic mean of the two quartiles.
• Trimean: The weighted arithmetic mean of the median and two quartiles.
• Winsorized mean: An arithmetic mean in which extreme values are replaced by values closer to the median.

Note that measures of central tendency are applied according to different levels of measures (type of a variable).

The best measure to use depends on the characteristics of your data and the specific question you’re trying to answer.

In summary, measures of central tendencies are fundamental tools in statistics whose use depends on the characteristics of the data being studied. The measures are used to summarize the data and are used to provide insight and foundation for further analysis. They also help in getting valuable insights for decision-making and prediction. Therefore, understanding the measures of central tendencies is essential to effectively analyzing and interpreting data.

FAQS about Measure of Central Tendency

1. Define the measure of central tendency.
2. What conditions must a measure of tendency follow?
3. Name widely used measures of central tendency.
4. What is the functionality of the measure of central tendencies?
5. What statistical measures can be applied on which level of measurement?

Reference

1) Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-
2) https://en.wikipedia.org/wiki/Central_tendency
3) Dodge, Y. (2005) The Concise Encyclopedia of Statistics. Springer,

R and Data Analysis

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The median is one of the three main measures of central tendency, alongside the mean and mode. It represents the middle value of an ordered dataset. It is a powerful and reliable summary statistic and widely used, especially in real-life scenarios where data is skewed or contains outliers. Unlike the mean, the median is not affected by extreme values, which makes it incredibly useful in various fields. For the formula of the median, read the post: formula of median and definition.

When the Median is Preferred over the Mean

Question: What is a measure of central tendency, and what are the common measures of central tendency? Also, when is the median preferred over the mean?

A measure of central tendency is the single numerical value considered most typical of the values of a quantitative variable.

The most common measure of central tendency is the mode (i.e., the most frequently occurring number)

The median (i.e., the middle point or fiftieth percentile), and the mean (i.e., the arithmetic average).

The median is preferred over the mean when the numbers are highly skewed (i.e., non-normally distributed).

Importance of Measures of Central Tendencies

Since measures of central tendency condense a bunch of information into a single, digestible value that represents the center of the data, this makes measures of central tendencies important for several reasons:

• Summarizing data: Instead of listing every data point, one can use a central tendency measure to get a quick idea of what is typical in the data set.
• Comparisons: By computing central tendency measures for different groups or datasets, one can easily compare them to see if there are any differences.
• Decision making: Central tendency measures can help to make wise decisions. For instance, knowing the average income in an area can help set prices. Imagine an organization is analyzing customer purchases. Knowing the average amount spent can help them tailor promotions or target specific customer groups.
• Identifying trends: Measures of central tendencies may help in observing the trend over time. This can be done by using different visualizations to see if there are any trends, like a rise in average house prices.

However, it is very important to understand these Measures of Central Tendency (mean, median, mode). Each measure of central tendency has its strengths and weaknesses. Choosing the right measure of central tendency depends on the kind of data and what one’s interest is to extract from and try to understand.

Real-Life Examples and Uses of Median

• Income & Salaries: The Median is used to represent the average income of a population more accurately. It is because A few ultra-rich individuals can skew the mean income upward. The median gives a more realistic picture of what a typical person earns. Example: If most people earn around $40,000–$60,000, but a few CEOs earn $10 million or more, the median income might be $55,000 while the mean income could be $95,000 — misleading!
• Education (Test/ Exame Scores): The median can be used to summarize exam results or performance data. A few very low or very high scores can distort the mean. For example, if most students score between 70 and 90, but a few score 10 or 100, the measure of central tendency, the median score, gives a better sense of central performance.
• Real Estate (Home Prices): Reporting the median home price is common in real estate. Why Median? It avoids distortion from a few very expensive or very cheap homes. For example, A city may have a median home price of $350,000, even if some luxury homes cost $5 million.
• Sports (Player Performance): To report median stats like race times, goals scored, or batting averages. To avoid skewed data from one amazing or terrible performance. For example, a runner’s median race time over 10 races can better reflect consistency.
• Healthcare (Medical Test Results): Reporting the median wait time in hospitals or median survival time in clinical trials may be beneficial. This is because medical data often contains outliers or skewed distributions. For example, if most patients wait 30 minutes, but a few wait 5 hours, the measure of central tendency, the median wait time, might be 35 minutes, while the mean could be misleadingly high.
• Customer Feedback (Review Rating): Median star rating for products or services. Filters out extremely negative or overly positive outliers. For example, if ratings are 1, 5, 5, 5, and 1, the mean is 3.4 but the median is 5, better reflecting the typical rating.
• Transportation (Travel Times): Apps like Google Maps or Waze often use median travel times to reflect a more realistic average, ignoring rare traffic jams or super fast times. For example, the median commute time may be 25 minutes, even if a few people experience 60-minute delays.

Summary

Scenario/ Use Case	Variable	Why Median should be used
Income reports	Salary	Avoids distortion by billionaires
House prices	Real estate values	Neutralizes luxury properties
ER performance	Patient wait times	Filters extreme delays
Test scores	Exam performance	Reduces skew from outliers
Travel times	Commute estimates	Reflects normal travel conditions
Product reviews	User ratings	Balances biased reviews

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