Mean is an essential concept in mathematics and statistics. The mean is the average or the most common value in a collection of numbers.
In statistics, it is a measure of central tendency of a probability distribution along median and mode. It is also referred to as an expected value.
It is a statistical concept that carries a major significance in finance.FinanceCFI's Finance Articles are designed as selfstudy guides to learn important finance concepts online at your own pace.
Browse hundreds of articles! The concept is used in various financial fields, including but not limited to portfolio managementPrivate Wealth ManagementPrivate wealth management is an investment practice that involves financial planning, tax management, asset protection and other financial services for high net worth individuals (HNWI) or accredited investors. Private wealth managers create a close working relationship with wealthy clients to help build a portfolio that achieves the client’s financial goals. and business valuationBusiness Valuation GlossaryThis business valuation glossary covers the most important concepts to know in valuing a company. This guide is part of CFI's Business Valuation Modeling.
Learn more about other mathematical concepts with CFI’s Math for Corporate Finance Course.
How to calculate mean?
There are different ways of measuring the central tendency of a set of values. There are multiple ways to calculate the mean. Here are the two most popular ones:
Arithmetic mean is the total of the sum of all values in a collection of numbers divided by the number of numbers in a collection. It is calculated in the following way:
In finance, the arithmetic mean may be misleading in the calculations of returns, as it does not consider the effects of volatility and compounding, producing an inflated value for the central point of the distribution.
Geometric mean is an nth root of the product of all numbers in a collection.
The formula for the geometric meanGeometric MeanThe geometric mean is the average growth of an investment computed by multiplying n variables and then taking the n square root.
In other words, it is the average return of an investment over time, a metric used to evaluate the performance of an investment portfolio.The geometric mean formula can be broken down to show is:
The geometric mean includes the volatility and compounding effects of returns. Thus, the geometric average provides a more accurate calculation of an average return.
Arithmetic mean example
Jim wants to find a stockStockWhat is a stock? An individual who owns stock in a company is called a shareholder and is eligible to claim part of the company’s residual assets and earnings (should the company ever be dissolved). The terms “stock”, “shares”, and “equity” are used interchangeably.
for investment. He is a big fan of Apple Inc. He knows that the company has strong financials. However, to ensure that this investment will bring him a substantial return, he has decided to check how the stock performed in the past.
He decides to find the average price of Apple’s share price for the past five months.
He gathered the monthly company’s stock prices from January 2018 to June 2018 and found the monthly returns. The stock prices and returns are summarized in the table below:
The formula used for the calculation would be the following:
Geometric mean example
In order to check the obtained result, Jim has decided to calculate the geometric mean return of Apple’s share price. However, it should be calculated not in percentages but in decimal numbers.
The geometric mean is equal to:
Additional resources
What You Need to Do in Order to Calculate the Mean, Median, or Mode
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Students often find that it is easy to confuse the mean, median, and mode. While all are measures of central tendency, there are important differences in what each one means and how they are calculated. Explore some useful tips to help you distinguish between the mean, median, and mode and learn how to calculate each measure correctly.
In order to understand the differences between the mean, median, and mode, start by defining the terms.
 The mean is the arithmetic average of a set of given numbers.
 The median is the middle score in a set of given numbers.
 The mode is the most frequently occurring score in a set of given numbers.
The mean, or average, is calculated by adding up the scores and dividing the total by the number of scores. Consider the following number set: 3, 4, 6, 6, 8, 9, 11. The mean is calculated in the following manner:
 3 + 4 + 6 + 6 + 8 + 9 + 11 = 47
 47 / 7 = 6.7
 The mean (average) of the number set is 6.7.
The median is the middle score of a distribution. To calculate the median
 Arrange your numbers in numerical order.
 Count how many numbers you have.
 If you have an odd number, divide by 2 and round up to get the position of the median number.
 If you have an even number, divide by 2. Go to the number in that position and average it with the number in the next higher position to get the median.
Consider this set of numbers: 5, 7, 9, 9, 11. Since you have an odd number of scores, the median would be 9. You have five numbers, so you divide 5 by 2 to get 2.5, and round up to 3. The number in the third position is the median.
What happens when you have an even number of scores so there is no single middle score? Consider this set of numbers: 1, 2, 2, 4, 5, 7. Since there is an even number of scores, you need to take the average of the middle two scores, calculating their mean.
Remember, the mean is calculated by adding the scores together and then dividing by the number of scores you added. In this case, the mean would be 2 + 4 (add the two middle numbers), which equals 6. Then, you take 6 and divide it by 2 (the total number of scores you added together), which equals 3. So, for this example, the median is 3.
Since the mode is the most frequently occurring score in a distribution, simply select the most common score as your mode. Consider the following number distribution of 2, 3, 6, 3, 7, 5, 1, 2, 3, 9.
The mode of these numbers would be 3 since three is the most frequently occurring number.
In cases where you have a very large number of scores, creating a frequency distribution can be helpful in determining the mode.
In some number sets, there may actually be two modes. This is known as bimodal distribution and it occurs when there are two numbers that are tied in frequency. For example, consider the following set of numbers: 13, 17, 20, 20, 21, 23, 23, 26, 29, 30. In this set, both 20 and 23 occur twice.
If no number in a set occurs more than once, then there is no mode for that set of data.
How do you determine whether to use the mean, median or mode? Each measure of central tendency has its own strengths and weaknesses, so the one you choose to use may depend largely on the unique situation and how you are trying to express your data.
 The mean utilizes all numbers in a set to express the measure of central tendency; however, outliers can distort the overall measure. For example, a couple of extremely high scores can skew the mean so that the average score appears much higher than most of the scores actually are.
 The median gets rid of disproportionately high or low scores, but it may not adequately represent the full set of numbers.
 The mode may be less influenced by outliers and is good at representing what is “typical” for a given group of numbers, but may be less useful in cases where no number occurs more than once.
Imagine a situation where a real estate agent wants a measure of the central tendency of homes she has sold in the last year. She makes a list of all of the totals:
 $75,000
 $75,000
 $150,000
 $155,000
 $165,000
 $203,000
 $750,000
 $755,000
The mean for this group is $291,000, the median is $160,000 and the mode is $75,000.
Which would you say is the best measure of central tendency of the set of sales numbers? If she wants the highest number, the mean is clearly the best option even though the total is skewed by the two very high numbers.
The mode, however, would not be a good choice because it is disproportionately low and not a good representation of her sales for the year. The median, on the other hand, seems to be a fairly good indicator of the “typical” sales prices of her real estate listings.
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Mean
The word mean, which is a homonym for multiple other words in the English language, is similarly ambiguous even in the area of mathematics. Depending on the context, whether mathematical or statistical, what is meant by the “mean” changes.
In its simplest mathematical definition regarding data sets, the mean used is the arithmetic mean, also referred to as mathematical expectation, or average. In this form, the mean refers to an intermediate value between a discrete set of numbers, namely, the sum of all values in the data set, divided by the total number of values.
The equation for calculating an arithmetic mean is virtually identical to that for calculating the statistical concepts of population and sample mean, with slight variations in the variables used:
The mean is often denoted as x̄, pronounced “x bar,” and even in other uses when the variable is not x, the bar notation is a common indicator of some form of mean.
In the specific case of the population mean, rather than using the variable x̄, the Greek symbol mu, or μ, is used. Similarly, or rather confusingly, the sample mean in statistics is often indicated with a capital X̄.
Given the data set 10, 2, 38, 23, 38, 23, 21, applying the summation above yields:

=  = 22.143 
As previously mentioned, this is one of the simplest definitions of the mean, and some others include the weighted arithmetic mean (which only differs in that certain values in the data set contribute more value than others), and geometric mean.
Proper understanding of given situations and contexts can often provide a person with the tools necessary to determine what statistically relevant method to use.
In general, mean, median, mode and range should ideally all be computed and analyzed for a given sample or data set since they elucidate different aspects of the given data, and if considered alone, can lead to misrepresentations of the data, as will be demonstrated in the following sections.
Median
The statistical concept of the median is a value that divides a data sample, population, or probability distribution into two halves. Finding the median essentially involves finding the value in a data sample that has a physical location between the rest of the numbers.
Note that when calculating the median of a finite list of numbers, the order of the data samples is important. Conventionally, the values are listed in ascending order, but there is no real reason that listing the values in descending order would provide different results.
In the case where the total number of values in a data sample is odd, the median is simply the number in the middle of the list of all values. When the data sample contains an even number of values, the median is the mean of the two middle values.
While this can be confusing, simply remember that even though the median sometimes involves the computation of a mean, when this case arises, it will involve only the two middle values, while a mean involves all the values in the data sample.
In the odd cases where there are only two data samples or there is an even number of samples where all the values are the same, the mean and median will be the same. Given the same data set as before, the median would be acquired in the following manner:
 2,10,21,23,23,38,38
 After listing the data in ascending order, and determining that there are an odd number of values, it is clear that 23 is the median given this case. If there were another value added to the data set:
 2,10,21,23,23,38,38,1027892
Since there are an even number of values, the median will be the average of the two middle numbers, in this case 23 and 23, the mean of which is 23.
Note that in this particular data set, the addition of an outlier (a value well outside the expected range of values), the value 1,027,892, has no real effect on the data set. If however the mean is computed for this data set, the result is 128,505.875.
This value is clearly not a good representation of the seven other values in the data set that are far smaller and closer in value than the average and the outlier. This is the main advantage of using the median in describing statistical data when compared to the mean.
While both, as well as other statistical values, should be calculated when describing data, if only one can be used, the median can provide a better estimate of a typical value in a given data set when there are extremely large variations between values.
Mode
In statistics, the mode is the value in a data set that has the highest number of recurrences. It is possible for a data set to be multimodal, meaning that it has more than one mode. For example:
2,10,21,23,23,38,38
Both 23 and 38 appear twice each, making them both a mode for the data set above.
How can I calculate the mean of two or three means?
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The standard statistical analysis of volcanic eruptions could be not very useful tools to describe temporal variations of the volcanic events clustering due to the shortness of the catalogues. A more appropriate approach is the multifractal analysis. If interevent times between eruptions are fractally distributed, they have a scale invariant struct…
How to Calculate the Mean
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In mathematics, the “mean” is a kind of average found by dividing the sum of a set of numbers by the count of numbers in the set.
[1] While it isn't the only kind of average, the mean is the one most people think of when speaking about an average.
You can use means for all kinds of useful purposes in your daily life, from calculating the time it takes you to get home from work, to working out how much money you spend in an average week.[2]

1
Determine the set of values you want to average. These numbers can be big or small, and there can be as many of them as you want.[3] Just make sure you are using real numbers and not variables.

2
Add your values together to find the sum. You can use a calculator, by hand, or a spreadsheet application to do so.[4]
 Example: 2+3+4+5+6=20{displaystyle 2+3+4+5+6=20}

3
Count the number of values in your group. Count all of the numbers added up. Identical values should still be counted, meaning if you have values that repeat in your set, each one still counts in determining your total. Do not include the sum (answer) of all the numbers added up when counting the quantity of the values.[5]
 Example: 2, 3, 4, 5, and 6 make for a total of five values.

4
Divide the sum of the set by the number of values. The result is the mean (a type of average) of your set. This implies that if each number in your set was the mean, they would add up to the same total.[6]
 Example: 20/5=4{displaystyle 20/5=4}. Therefore, 4 is the mean of the numbers. You can check your calculations by multiplying the mean by the number of values in the set. In this case, multiply 4{displaystyle 4} (the mean) by the 5{displaystyle 5} (the number of values in the set) and your result will be 20{displaystyle 20} (4∗5=20{displaystyle 4*5=20}).
Add New Question
 Question What do I do if my mean/average has a remainder? Convert the remainder to a decimal or fraction. For example: the average of 5, 12, and 17 is 34 ÷ 3, which is 11 with a remainder of 1. Convert the remainder to the fraction 1/3 or the decimal .33. Thus, the average is 111/3 or 11.33.
 Question How do I calculate the mode? It is the number in a set which appears most often. In the set {1 7 9 0 4 5 4}, 4 is the mode.
 Question How do I find the range? The range of a data set is the difference between the largest number and the smallest number in the set.
 Question What if there's a remainder? That's fine. A mean does not have to be a whole number.
 Question How do you find the mean with negative numbers? Add them algebraically, and divide by the number of numbers.
 Question How do I calculate the mean of 45, 12, 15, 92 and 61? Add them all together and divide the answer by five (since there are five numbers).
 Question How do I find the median? Add the highest number and the lowest number and divide by two.
 Question How do I calculate survey results? Assuming the survey responses consist of numbers, add the numbers together, and divide by the number of responses.
 Question Can a mean be a decimal point? Yes.
 Question Does this work for all numbers? It works for all rational numbers.
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2. Mean and standard deviation
(pronounced “x bar”) signifies the mean; x is each of the values of urinary lead; n is the number of these values; and σ , the Greek capital sigma (our “S”) denotes “sum of”. A major disadvantage of the mean is that it is sensitive to outlying points. For example, replacing 2.2 by 22 in Table 1.1 increases the mean to 2.82 , whereas the median will be unchanged.
As well as measures of location we need measures of how variable the data are. We met two of these measures, the range and interquartile range, in Chapter 1.
The range is an important measurement, for figures at the top and bottom of it denote the findings furthest removed from the generality. However, they do not give much indication of the spread of observations about the mean. This is where the standard deviation (SD) comes in.
The theoretical basis of the standard deviation is complex and need not trouble the ordinary user. We will discuss sampling and populations in Chapter 3.
A practical point to note here is that, when the population from which the data arise have a distribution that is approximately “Normal” (or Gaussian), then the standard deviation provides a useful basis for interpreting the data in terms of probability.
The Normal distribution is represented by a family of curves defined uniquely by two parameters, which are the mean and the standard deviation of the population.
The curves are always symmetrically bell shaped, but the extent to which the bell is compressed or flattened out depends on the standard deviation of the population.
However, the mere fact that a curve is bell shaped does not mean that it represents a Normal distribution, because other distributions may have a similar sort of shape.
Many biological characteristics conform to a Normal distribution closely enough for it to be commonly used – for example, heights of adult men and women, blood pressures in a healthy population, random errors in many types of laboratory measurements and biochemical data.
Figure 2.1 shows a Normal curve calculated from the diastolic blood pressures of 500 men, mean 82 mmHg, standard deviation 10 mmHg. The ranges representing [+1SD, +12SD, and +3SD] about the mean are marked.
A more extensive set of values is given in Table A of the print edition.
Figure 2.1
The reason why the standard deviation is such a useful measure of the scatter of the observations is this: if the observations follow a Normal distribution, a range covered by one standard deviation above the mean and one standard deviation below it
includes about 68% of the observations; a range of two standard deviations above and two below () about 95% of the observations; and of three standard deviations above and three below () about 99.7% of the observations. Consequently, if we know the mean and standard deviation of a set of observations, we can obtain some useful information by simple arithmetic. By putting one, two, or three standard deviations above and below the mean we can estimate the ranges that would be expected to include about 68%, 95%, and 99.7% of the observations.
Standard deviation from ungrouped data
The standard deviation is a summary measure of the differences of each observation from the mean. If the differences themselves were added up, the positive would exactly balance the negative and so their sum would be zero. Consequently the squares of the differences are added.
The sum of the squares is then divided by the number of observations minus oneto give the mean of the squares, and the square root is taken to bring the measurements back to the units we started with. (The division by the number of observations minus oneinstead of the number of observations itself to obtain the mean square is because “degrees of freedom” must be used.
In these circumstances they are one less than the total. The theoretical justification for this need not trouble the user in practice.)
To gain an intuitive feel for degrees of freedom, consider choosing a chocolate from a box of n chocolates. Every time we come to choose a chocolate we have a choice, until we come to the last one (normally one with a nut in it!), and then we have no choice. Thus we have n1 choices, or “degrees of freedom”.
The calculation of the variance is illustrated in Table 2.1 with the 15 readings in the preliminary study of urinary lead concentrations (Table 1.2). The readings are set out in column (1).
In column (2) the difference between each reading and the mean is recorded. The sum of the differences is 0.
In column (3) the differences are squared, and the sum of those squares is given at the bottom of the column.
Table 2.1
The sum of the squares of the differences (or deviations) from the mean, 9.96, is now divided by the total number of observation minus one, to give the variance.Thus,In this case we find:Finally, the square root of the variance provides the standard deviation:
from which we get
This procedure illustrates the structure of the standard deviation, in particular that the two extreme values 0.1 and 3.2 contribute most to the sum of the differences squared.
Calculator procedure
Most inexpensive calculators have procedures that enable one to calculate the mean and standard deviations directly, using the “SD” mode. For example, on modern Casio calculators one presses SHIFT and '.' and a little “SD” symbol should appear on the display.
On earlier Casios one presses INV and MODE , whereas on a Sharp 2nd F and Stat should be used. The data are stored via the M+ button. Thus, having set the calculator into the “SD” or “Stat” mode, from Table 2.1 we enter 0.1 M+ , 0.4 M+ , etc.
When all the data are entered, we can check that the correct number of observations have been included by Shift and n, and “15” should be displayed. The mean is displayed by Shift and and the standard deviation by Shift and . Avoid pressing Shift and AC between these operations as this clears the statistical memory. There is another button on many calculators.
This uses the divisor n rather than n – 1 in the calculation of the standard deviation. On a Sharp calculator is denoted , whereas is denoted s.
These are the “population” values, and are derived assuming that an entire population is available or that interest focuses solely on the data in hand, and the results are not going to be generalised (see Chapter 3 for details of samples and populations).
As this situation very rarely arises, should be used and ignored, although even for moderate sample sizes the difference is going to be small. Remember to return to normal mode before resuming calculations because many of the usual functions are not available in “Stat” mode. On a modern Casio this is Shift 0. On earlier Casios and on Sharps one repeats the sequence that call up the “Stat” mode. Some calculators stay in “Stat” mode even when switched off.Mullee (1) provides advice on choosing and using a calculator. The calculator formulas use the relationship
 The right hand expression can be easily memorised by the expression mean of the squares minus the mean square”. The sample variance is obtained from
The above equation can be seen to be true in Table 2.1, where the sum of the square of the observations, , is given as 43.7l. We thus obtain
the same value given for the total in column (3). Care should be taken because this formula involves subtracting two large numbers to get a small one, and can lead to incorrect results if the numbers are very large.
For example, try finding the standard deviation of 100001, 100002, 100003 on a calculator. The correct answer is 1, but many calculators will give 0 because of rounding error.
The solution is to subtract a large number from each of the observations (say 100000) and calculate the standard deviation on the remainders, namely 1, 2 and 3.
Standard deviation from grouped data
We can also calculate a standard deviation for discrete quantitative variables.
For example, in addition to studying the lead concentration in the urine of 140 children, the paediatrician asked how often each of them had been examined by a doctor during the year. After collecting the information he tabulated the data shown in Table 2.
2 columns (1) and (2). The mean is calculated by multiplying column (1) by column (2), adding the products, and dividing by the total number of observations. Table 2.2
As we did for continuous data, to calculate the standard deviation we square each of the observations in turn. In this case the observation is the number of visits, but because we have several children in each class, shown in column (2), each squared number (column (4)), must be multiplied by the number of children.
The sum of squares is given at the foot of column (5), namely 1697. We then use the calculator formula to find the variance:and .Note that although the number of visits is not Normally distributed, the distribution is reasonably symmetrical about the mean.
The approximate 95% range is given byThis excludes two children with no visits and six children with six or more visits. Thus there are eight of 140 = 5.7% outside the theoretical 95% range.Note that it is common for discrete quantitative variables to have what is known as skeweddistributions, that is they are not symmetrical.
One clue to lack of symmetry from derived statistics is when the mean and the median differ considerably. Another is when the standard deviation is of the same order of magnitude as the mean, but the observations must be nonnegative. Sometimes a transformation will convert a skewed distribution into a symmetrical one.
When the data are counts, such as number of visits to a doctor, often the square root transformation will help, and if there are no zero or negative values a logarithmic transformation will render the distribution more symmetrical.
Data transformation
An anaesthetist measures the pain of a procedure using a 100 mm visual analogue scale on seven patients. The results are given in Table 2.3, together with the log etransformation (the ln button on a calculator). Table 2.3
The data are plotted in Figure 2.2, which shows that the outlier does not appear so extreme in the logged data. The mean and median are 10.29 and 2, respectively, for the original data, with a standard deviation of 20.22.
Where the mean is bigger than the median, the distribution is positively skewed. For the logged data the mean and median are 1.24 and 1.10 respectively, indicating that the logged data have a more symmetrical distribution.
Thus it would be better to analyse the logged transformed data in statistical tests than using the original scale.Figure 2.2
In reporting these results, the median of the raw data would be given, but it should be explained that the statistical test wascarried out on the transformed data. Note that the median of the logged data is the same as the log of the median of the raw data – however, this is not true for the mean.
The mean of the logged data is not necessarily equal to the log of the mean of the raw data. The antilog (exp or on a calculator) of the mean of the logged data is known as the geometric mean,and is often a better summary statistic than the mean for data from positively skewed distributions.
For these data the geometric mean in 3.45 mm.
Between subjects and within subjects standard deviation
Need to Know Your GPA? Calculate the Mean or Average
Given a list of numbers, it is easy to determine the arithmetic mean, or average. The average is simply the sum of the numbers in a given problem, divided by the number of numbers added together. For example, if four number are added together their sum is divided by four to find the average or arithmetic mean.
Average or arithmetic mean is sometimes confused with two other concepts: mode and median. The mode is the most frequent value in a set of numbers, while the median is the number in the middle of the range of a given set.
It's important to know how to calculate the mean or average of a set of numbers. Among other things, this will allow you to calculate your grade point average. However, you'll need to calculate the mean for several other situations, too.
The concept of an average allows statisticians, demographers, economists, biologists, and other researchers to better understand the most common situations.
For example, by determining the average income of an American family and comparing it to the average cost of a home, it's possible to better understand the magnitude of economic challenges facing most American families.
Similarly, by looking at the average temperature in a particular area at a particular time of year, it's possible to predict the probable weather and make a wide range of decisions appropriately.
While averages can be very useful tools, they can also be misleading for a variety of reasons. In particular, averages can obscure the information contained in data sets. Here are a few examples of how averages can be misleading:
 John's grades include a 4.5 in math, a 4.0 in science, a 2.0 in English and a 2.5 in History. After averaging his scores, his advisor decided that John is a straight “B” student. In fact, however, John is quite talented in math and science and needs remediation in English and history.
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