Arithmetic Mean – Tutorial With Examples

22.12.22 Measures of central tendency Time to read: 6min

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Arithmetic-mean-Definition

The arithmetic mean, also known as the “mean” or “average”, is a fundamental measure of central tendency in statistics. It defines the sum of all the numbers in a group divided by the total number of items in the sequence or data set. In addition, it serves as a critical tool for interpreting and understanding data in a research study.  This article will analyze the arithmetic mean and how it is calculated with tutorials and examples.

Arithmetic Mean – In a Nutshell

  • The ratio of the total number of observations to the sum of all observations is known as the arithmetic mean.
  • The arithmetic mean can explain or describe ideas unrelated to statistics.
  • The arithmetic mean can be compared to a center of gravity in physical terms.
  • The average deviation of a data set’s data points from the mean is the standard deviation.

Definition: Arithmetic mean

The term “arithmetic mean” refers to a value determined by dividing the total number of values in a set by the sum of its members. Knowing the distinctions between medianmean, and mode is a prerequisite to understanding arithmetic mean.

A dataset’s mean (average) is calculated by summing all the numbers in the set and then dividing by the total number of values in the set. When a data collection is ranked from least to greatest, the median is the midpoint, while the number that appears most frequently in a data set is called the mode.

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Arithmetic mean formulas

A sample is a particular group from which you will gather data, whereas a population is an entire group from which you intend to conclude. The sample size is always smaller than the population as a whole.

The sample and population mean are two different averages used in statistics. Only a few observations—selected from the population data—are considered for calculating the sample mean. On the other hand, the arithmetic mean can be used when the population mean computes the average value by considering all the population’s observations.

Population mean

A population mean is a ratio of the sum of the values to the number of values. Every component from the possible set of observations is included in the population mean and is an effective use of arithmetic mean.

Formula Explanation
= Population mean

= Sum of all items

= Total number of items

Sample mean

The central tendency, standard deviation, and variance of a collection of data can all be determined using the sample mean. Calculating population averages is just one of the many uses for the sample mean.

Formula Explanation
= Sample mean

= Sum of all values

= Number of terms

Calculating the arithmetic mean

Imagine that you were interested in learning about the weather in Shimla. On the internet, you can find:

  • The temperatures for many days
  • Information on the temperature in the past and present
  • Forecasts for the temperature in the future

Researchers chose to utilize representative values that could account for a wide range of data in place of this lengthy list. We describe the weather over about a month using terminology like arithmetic mean, median, and mode rather than the weather for each specific day.

Example

Step 1: Find the sum of all the observations.

Step 2: Multiply the frequency with its corresponding value and add them. This step is applicable only in the case of discrete and continuous series.

Step 3: Find the number of observations. However, in the case of discrete and continuous series, we add up all the frequencies.

Step 4: Divide the result in Step 1 or Step 2 (as the case may be) with the result in

Step 5: The resultant figure is the mean.

Example

You want to find out how much coffee your colleagues drink each day on average.

Dataset: 1 2 1 2 1 0 3 1 0 3

Step 1: Find the sum of all values

Step 2: Divide by the number of values

You must note that the number 0 is included as a value in the data set whenever calculating the arithmetic mean.

The outlier effect on the arithmetic mean

Outliers are numbers in a data set that is vastly larger or smaller than the other values in the set.

Outliers, such as the mean, can have a disproportionate effect on statistical results, which can result in misleading interpretations of the arithmetic mean.

Example

A data set includes the values: 1, 2, 3, and 34.

Step 1: Find the sum of all values

Step 2: Divide by the number of values

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 during calculation if the arithmetic mean.

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Mean, median, and mode in an arithmetic mean

Continuous variables are often associated with something we can measure, while discrete variables are typically associated with something we can count. Some variables have both quantitative and categorical options. The classification of data relies on the purpose of gathering it.

1. Qualitative variables, often known as categorical variables, can be categorized according to certain traits or features by naming the categories of this variable (whether with words or numerals). When asked questions like What kind of advertising do you use?” they often give descriptive answers:

  • There may be just two possible values (like “yes” or “no”).
  • Might be a number, such as a zip code.
  • This variable’s averages cannot be found.

 

2. Quantitative variables (Numerical variable):

When the values of a variable are measured, qualitative numerical variables, also known as categorical variables, may be grouped into several groups based on certain traits or attributes. All that this variable does is list the categories (whether with words or numerals). Thereby, the arithmetic mean comes as a result of descriptive answers to inquiries like “What kind of advertising do you use?”:

  • Might only allow for two possible values (like “yes” or “no”).
  • Might be a number, like a zip code.
  • For this variable, no averages could be found.

3. Discrete variables (Quantitative):

They presume countable values. It can take on several different values.

Examples

  • Number of kids in a family
  • Car crash frequency
  • Shoe sizes

4. Continuous variables (Quantitative):

They can assume two specific values between an infinite number of other values. Decimals and fractions are frequently used in them.

Examples

  • Weather
  • Rain
  • Gasoline

Distribution shapes

The mean and median have the same value in a normal distribution, however, in a skewed distribution, they have distinct values:

The mean will be located to the left of the median in a left-skewed, negative distribution. The mean will be to the right of the median in a right-skewed, positive distribution.

Arithmetic-mean-positively-skewed-distribution
Arithmetic-mean-negatively-skewed-distribution

FAQs

  • In sports like cricket, the arithmetic mean is utilized to figure out the typical score.
  • It is also employed in various disciplines, including anthropology, history, and economics.
  • To gauge global warming, the world’s average temperature is also measured using the arithmetic mean.
  • It also calculates how much rain falls in a specific area each year.

Because it considers every value in the data set, the arithmetic mean, also known as mean, is regarded as the best measure.

The mean value will vary if any value in the data set changes, but neither the median nor the mode will be affected.

The value of each item in a series, including the massive and very small ones, is considered by the arithmetic mean.

As a result, only the arithmetic mean is impacted by outlier values in the series.