Following your probability distribution table, enter each possible value x and its probability p, one row at a time. The light gray example is the prize from one raffle ticket ($0 with 70%, $1 with 20%, $5 with 9% and $100 with 1%).
Enter numbers only. A probability can be entered as 0.2, 20% or 1/5, and the probabilities in all rows must add up to 1. Values can also be negative (a loss). Leave both "value" and "probability" blank in rows you do not use.
Result and graph
Enter pairs of values and probabilities in the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and graph
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter a probability distribution table (pairs of a possible value \(x\) and its probability \(p\), 2 to 10 rows), and you get the expected value \(E(X)\), the variance \(\mathrm{Var}(X)\) and the standard deviation \(\sigma(X)\) on the spot
- Answers are shown both as a fraction in lowest terms (the exact value), such as \(\dfrac{7}{2}\), and as a decimal. You can also see the steps, with your numbers put into \(E(X) = x_1 p_1 + x_2 p_2 + \cdots\)
- It automatically checks that the probabilities add up to exactly 1. If they do not, it tells you what the current total is
- You can enter a probability as 0.2, 20% or 1/5. Values can also be negative (a loss)
- The result includes a bar chart of the probability distribution, with the expected value \(E(X)\) marked by a vertical dashed line
This page handles random variables with a finite number of possible values (discrete random variables). It does not handle random variables that take continuous values, such as height.
What is this calculation used for?
Putting a number on "what you get back per ticket" in lotteries and loot boxes
The expected value, "prize × chance of winning" added up over every prize, is exactly the average return per ticket. In big US lottery games such as Powerball, about half of ticket sales goes into the prize pool, so the expected value of a ticket is well below its price.
You can do the same calculation for loot boxes in video games. Using the published drop rates (probabilities), it is the basis for estimating "about how much it will cost to get the item you want".
Setting insurance premiums (the basis of how insurers set prices)
Life and auto insurance premiums start from the expected amount paid out: "each claim payment × the chance of that accident or illness", added up. Operating costs and other amounts are then added on top. There is a whole profession, the actuary, that specializes in this kind of calculation.
From the customer's side, insurance is a deal where you pay more than you get back on average (by expected value). What you buy is peace of mind against a rare but large loss.
Comparing investment return and risk on the same scale
In investing, it is common to treat the expected value of the return (rate of return) as "the gain you can expect" and the standard deviation as "the risk (the size of the swings)". If two investments have the same expected return, the one with the smaller standard deviation moves up and down more gently.
The idea on this page, looking at "how high the expected value is" and "how big the spread is" as two separate numbers, carries straight over into real-world finance.
Game balancing (designing rewards and damage)
Board game and video game designers calculate expected values, such as "the sum of two dice", "the damage per attack" and "the value of an item from a treasure chest", to balance strength and rewards.
For example, "100 damage with a 50% chance" and "always 50 damage" both have an expected value of 50, but their variances differ, so the two moves feel different to play. Designers tune both the expected value and the spread to create the experience they want.
Formulas and graph
Expected value (mean of a random variable)
Graph
Standard notation
(the usual math form)
\(E(X)\)
\(=\)
\(x_1\)
\(\times\)
\(p_1\)
\(+\)
\(\cdots\)
\(+\)
\(x_n\)
\(\times\)
\(p_n\)
In words
(symbols replaced with words)
③
\(E(X)\): expected value
\(=\)
①
\(x_1\): first value
\(\times\)
②
\(p_1\): its probability
\(+\)
\(\cdots\)
\(+\)
\(x_n\): last value
\(\times\)
\(p_n\): its probability
The formula in words
①
Take each of the
\(x_1,\ x_2,\ \dots,\ x_n\): possible values
②
multiply it by its
\(p_1,\ p_2,\ \dots,\ p_n\): probabilities of those values
and add them all up
③
and you get the
\(E(X)\): expected value
Quick example
The expected value of the number \(X\) you get when you roll a die once (each number has probability \(\dfrac{1}{6}\)) is
\(E(X)\): expected roll
\(=\)
roll 1
\(\times\)
probability \(\frac{1}{6}\)
\(+\)
\(\cdots\)
\(+\)
roll 6
\(\times\)
probability \(\frac{1}{6}\)
\(E(X) = 1 \times \dfrac{1}{6} + 2 \times \dfrac{1}{6} + 3 \times \dfrac{1}{6} + 4 \times \dfrac{1}{6} + 5 \times \dfrac{1}{6} + 6 \times \dfrac{1}{6}\)
\(E(X) = \dfrac{1 + 2 + 3 + 4 + 5 + 6}{6} = \dfrac{21}{6} = \dfrac{7}{2} = 3.5\)
Key idea
The expected value tells you "on average, how much you get per try". It is an average weighted by probability (the sum of each value × how likely it is). If you roll a die thousands or tens of thousands of times and average the numbers you get, the average gets closer and closer to this 3.5 (the law of large numbers).
You can never actually roll a 3.5. The key is to think of the expected value not as "a prediction of the next result" but as "the average per try when you repeat many times".
As the graph above shows, if you think of the probability distribution as weights placed along a line, the expected value is the balance point.
Variance (how spread out the values are)
Standard notation
(the usual math form)
\(\mathrm{Var}(X)\)
\(=\)
\(E(X^{2})\)
\(-\)
\([E(X)]^{2}\)
In words
(symbols replaced with words)
③
\(\mathrm{Var}(X)\): variance
\(=\)
①
\(E(X^2)\): mean of the squares
\(-\)
②
\([E(X)]^2\): square of the expected value
The formula in words
①
Take the
\(E(X^2)\): square first, then average
②
subtract the
\([E(X)]^2\): average first, then square
③
and you get the
\(\mathrm{Var}(X)\): variance
Quick example
The variance of the number \(X\) you roll on a die (expected value \(E(X) = \dfrac{7}{2}\)) is
\(\mathrm{Var}(X)\): variance of the roll
\(=\)
\(E(X^2)\): mean of the squared rolls
\(-\)
square of the expected value \(\left(\frac{7}{2}\right)^2\)
\(E(X^{2}) = 1^{2} \times \dfrac{1}{6} + 2^{2} \times \dfrac{1}{6} + 3^{2} \times \dfrac{1}{6} + 4^{2} \times \dfrac{1}{6} + 5^{2} \times \dfrac{1}{6} + 6^{2} \times \dfrac{1}{6} = \dfrac{91}{6}\)
\(\mathrm{Var}(X) = \dfrac{91}{6} - \left(\dfrac{7}{2}\right)^{2} = \dfrac{182}{12} - \dfrac{147}{12} = \dfrac{35}{12}\)
Key idea
The variance tells you how spread out the values are around the expected value. Its original definition squares each distance from the expected value \(\mu = E(X)\) (the deviation) and averages them: \(\mathrm{Var}(X) = (x_1 - \mu)^2 p_1 + (x_2 - \mu)^2 p_2 + \cdots + (x_n - \mu)^2 p_n\). You can read the meaning, "the average of the squared deviations", right off this formula.
However, the definition needs one subtraction from \(\mu\) for every row, which is a lot of work. Rearranging it gives the simpler form above, \(\mathrm{Var}(X) = E(X^2) - [E(X)]^2\) ("the mean of the squares minus the square of the mean"), often called the shortcut formula. Textbooks use it for most actual calculations, and this calculator also shows its steps with this formula.
Even with the same expected value, a larger variance means a distribution with bigger swings between winning and losing. For example, "a ticket that always pays $100" and "a ticket that pays $0 or $200 with equal chance" both have an expected value of $100, but the second has a larger variance.
Standard deviation
Standard notation
(the usual math form)
In words
(symbols replaced with words)
\(\sigma(X)\)
\(=\)
\(\sqrt{\mathrm{Var}(X)}\)
②
\(\sigma(X)\): standard deviation
\(=\)
①
positive square root of the variance \(\mathrm{Var}(X)\)
The formula in words
①
Take the positive square root of the
\(\mathrm{Var}(X)\): variance
②
and you get the
\(\sigma(X)\): standard deviation
Quick example
The standard deviation of the number \(X\) you roll on a die (variance \(\mathrm{Var}(X) = \dfrac{35}{12}\)) is
\(\sigma(X)\): standard deviation of the roll
\(=\)
positive square root of the variance \(\frac{35}{12}\)
\(\sigma(X) = \sqrt{\dfrac{35}{12}} \approx 1.71\)
Key idea
The variance is the average of "squared deviations", so its units are squared too (if the prize is in dollars, the variance is in dollars × dollars). That makes it hard to get a feel for the size of the spread, so you take the square root to return to the original units. That is the standard deviation.
When you look at it together with the expected value, as in "a raffle ticket with an expected value of $1.65 and a standard deviation of about $9.99", you can compare "how big the swings around the average are" in the same units.
The expected value \(E(X) = x_1 p_1 + x_2 p_2 + \cdots + x_n p_n\) is an average weighted by probability: add up "each value × its probability". The spread is measured by the variance \(\mathrm{Var}(X) = E(X^2) - [E(X)]^2\) and its square root, the standard deviation \(\sigma(X)\).
Symbols and terms
Symbols
| \(X\) | capital X | A random variable: a variable whose value is decided by the result of a trial, such as the number you roll on a die. Because it is a "container for a result" whose content is not yet decided, it is written with a capital letter to tell it apart from ordinary variables. |
| \(x_1,\ x_2,\ \dots,\ x_n\) | x sub 1, x sub 2, ..., x sub n | Each of the possible values of the random variable \(X\). The small number at the lower right (the subscript) only tells which value it is in the list; it has nothing to do with the value itself. |
| \(p_1,\ p_2,\ \dots,\ p_n\) | p sub 1, p sub 2, ..., p sub n | The probability of each value. \(p\) is the first letter of "probability". They always add up to 1. |
| \(n\) | n | The number of possible values (the number of rows in the probability distribution table). The letter \(n\), from "number", is often used for a count. |
| \(E(X)\) | E of X | The expected value of the random variable \(X\). \(E\) is the first letter of "expected value" (or "expectation"). It stands for one number, "the expected value of \(X\)"; it is not \(E\) times \(X\). |
| \(\mu\) | mu | The Greek letter often used for the expected value (the mean). It corresponds to "m" for "mean". Textbooks set \(\mu = E(X)\) to write the definition of the variance more compactly. |
| \(\mathrm{Var}(X)\) | variance of X | The variance of the random variable \(X\). It is the average of the squared distances from the expected value (the deviations) and shows how spread out the values are. Some books write it as \(V(X)\) or \(\sigma^2\). |
| \(\sigma(X)\) | sigma of X | The standard deviation of the random variable \(X\). \(\sigma\) is the lowercase Greek letter sigma, the Greek "s" for "standard deviation". It is the positive square root of the variance. Also written \(\sigma\) or SD(X). |
| \(X^2\) | X squared | A new random variable made by squaring the value of \(X\). \(E(X^2)\) is "square each possible value first, then take the average weighted by probability". |
| \(\sqrt{\phantom{0}}\) | square root (radical sign) | The symbol for the number (0 or more) that gives the number inside when squared, the positive square root. For example, \(\sqrt{9} = 3\). The standard deviation is found by putting the variance under this symbol. |
Terms
| random variable | A variable whose value is decided by the result of a trial (an experiment or observation whose result is decided by chance), such as the number rolled on a die or the prize from a raffle ticket. It is usually written with a capital \(X\). |
| probability distribution | The list that matches each possible value of a random variable with its probability. Written as a table, it is a probability distribution table, and the input fields on this page take that table as it is. |
| trial | An experiment or observation that can be repeated under the same conditions and whose result is decided by chance, such as rolling a die or drawing a raffle ticket. |
| expected value | The mean of a random variable. It is each value times its probability, all added up, and it approaches "the average per try" when the trial is repeated many times. The symbol is \(E(X)\) or \(\mu\). |
| weighted average | An average where each value is multiplied by a "weight" before adding, instead of simply adding and dividing by the count. The expected value is a weighted average with the probabilities as the weights. |
| variance | A measure of how spread out the values are. It squares the distances from the expected value (the deviations) and averages them, weighted by probability. The symbol is \(\mathrm{Var}(X)\). |
| standard deviation | The positive square root of the variance. Unlike the variance, it is in the same units as the original values, so it is easier to compare how spread out things are. The symbol is \(\sigma(X)\). |
| deviation | The distance of a value from the expected value, \(x - \mu\). Deviations are a mix of positive and negative numbers, and their plain average is always 0, so the variance squares them before averaging. |
| law of large numbers | The rule that as you repeat a trial more and more times, the average of the actual results gets closer and closer to the expected value. It is the basis for reading the expected value as "the average per try over many repetitions". |
| discrete random variable | A random variable whose possible values are separate points (a finite number of values, or whole numbers), like the numbers on a die (1 to 6). This page handles this type. A random variable that takes continuous values, such as height, is called a continuous random variable, and its expected value is found with an integral. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over the topics in this list is the fastest way forward.
| Basic probability (Grade 7 to high school) |
|
| Working with fractions and decimals (Grades 5–7) |
|
| The mean (Grade 6 to high school) |
|
| Squares and square roots (Grade 8) |
|
| Describing data (high school statistics) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the expected value E(X) (raffle example)
| Value x₁ (losing ticket, $0) | 0 |
| Probability p₁ | 0.7 |
| Value x₂ (prize $1) | 1 |
| Probability p₂ | 0.2 |
| Value x₃ (prize $5) | 5 |
| Probability p₃ | 0.09 |
| Value x₄ (prize $100) | 100 |
| Probability p₄ | 0.01 |
| Sum of probabilities (check that it is 1) | =B2+B4+B6+B8 |
| Expected value E(X) | =B1*B2+B3*B4+B5*B6+B7*B8 |
Table to find the variance Var(X) (raffle example)
| Value x₁ | 0 |
| Probability p₁ | 0.7 |
| Value x₂ | 1 |
| Probability p₂ | 0.2 |
| Value x₃ | 5 |
| Probability p₃ | 0.09 |
| Value x₄ | 100 |
| Probability p₄ | 0.01 |
| Expected value E(X) | =B1*B2+B3*B4+B5*B6+B7*B8 |
| Mean of the squares E(X²) | =B1^2*B2+B3^2*B4+B5^2*B6+B7^2*B8 |
| Variance Var(X) = E(X²) − [E(X)]² | =B10-B9^2 |
Table to find the standard deviation σ(X)
| Variance Var(X) | 99.7275 |
| Standard deviation σ(X) = √Var(X) | =SQRT(B1) |
After pasting, the upper rows (values and probabilities) are your inputs and the lower rows are calculated automatically. "*" is multiplication, "^2" means squared, and SQRT is the function for the square root.
The first table is the raffle example ($0 with 70%, $1 with 20%, $5 with 9% and $100 with 1%). The sum of the probabilities is 1 and the expected value is 1.65 (dollars).
The second table is the variance of the same raffle: the mean of the squares, 102.45, minus the square of the expected value, 2.7225, which is 99.7275.
The third table finds the standard deviation from that variance, about 9.99 (dollars). To add rows, add more value and probability pairs and extend the "B1*B2+…" part of the formulas in the same pattern.
The first table is the raffle example ($0 with 70%, $1 with 20%, $5 with 9% and $100 with 1%). The sum of the probabilities is 1 and the expected value is 1.65 (dollars).
The second table is the variance of the same raffle: the mean of the squares, 102.45, minus the square of the expected value, 2.7225, which is 99.7275.
The third table finds the standard deviation from that variance, about 9.99 (dollars). To add rows, add more value and probability pairs and extend the "B1*B2+…" part of the formulas in the same pattern.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the expected value E(X) (raffle example)
| Value x₁ (losing ticket, $0) | 0 |
| Probability p₁ | 0.7 |
| Value x₂ (prize $1) | 1 |
| Probability p₂ | 0.2 |
| Value x₃ (prize $5) | 5 |
| Probability p₃ | 0.09 |
| Value x₄ (prize $100) | 100 |
| Probability p₄ | 0.01 |
| Sum of probabilities (check that it is 1) | =B2+B4+B6+B8 |
| Expected value E(X) | =B1*B2+B3*B4+B5*B6+B7*B8 |
Table to find the variance Var(X) (raffle example)
| Value x₁ | 0 |
| Probability p₁ | 0.7 |
| Value x₂ | 1 |
| Probability p₂ | 0.2 |
| Value x₃ | 5 |
| Probability p₃ | 0.09 |
| Value x₄ | 100 |
| Probability p₄ | 0.01 |
| Expected value E(X) | =B1*B2+B3*B4+B5*B6+B7*B8 |
| Mean of the squares E(X²) | =B1^2*B2+B3^2*B4+B5^2*B6+B7^2*B8 |
| Variance Var(X) = E(X²) − [E(X)]² | =B10-B9^2 |
Table to find the standard deviation σ(X)
| Variance Var(X) | 99.7275 |
| Standard deviation σ(X) = √Var(X) | =SQRT(B1) |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the values and probabilities with your own numbers.
How to calculate it in Python
from fractions import Fraction
# Probability distribution table (value and probability pairs). Raffle example: prize and chance of winning it
# A fractional probability can be written like Fraction(1, 6)
values = [Fraction(0), Fraction(1), Fraction(5), Fraction(100)]
probabilities = [Fraction("0.7"), Fraction("0.2"), Fraction("0.09"), Fraction("0.01")]
# Check that the probabilities add up to 1
total = sum(probabilities)
if total != 1:
raise ValueError(f"The probabilities do not add up to 1 (total: {total})")
# Expected value E(X) = Σ x_i p_i
mean = sum(x * p for x, p in zip(values, probabilities))
# Mean of the squares E(X²) and variance Var(X) = E(X²) − [E(X)]²
mean_of_squares = sum(x * x * p for x, p in zip(values, probabilities))
variance = mean_of_squares - mean * mean
# Standard deviation σ(X) = √Var(X)
std_dev = float(variance) ** 0.5
print(f"Expected value E(X): {mean} = {float(mean)}")
print(f"Variance Var(X): {variance} = {float(variance)}")
print(f"Standard deviation σ(X): {std_dev}")
With the fractions module from the standard library, you can calculate with exact fractions and no decimal rounding errors. This example is the raffle ($0 with 70%, $1 with 20%, $5 with 9% and $100 with 1%). When you run it, it shows the expected value 33/20 = 1.65, the variance 39891/400 = 99.7275 and the standard deviation of about 9.9864. Change the values and probabilities and run it.
How to write it in LaTeX and other math languages (copy and paste)
Expected value (mean of a random variable)
E(X) = x₁p₁ + x₂p₂ + … + xₙpₙ
E(X) = x_1 p_1 + x_2 p_2 + \cdots + x_n p_n
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo>
<mo>=</mo>
<msub><mi>x</mi><mn>1</mn></msub><msub><mi>p</mi><mn>1</mn></msub>
<mo>+</mo>
<msub><mi>x</mi><mn>2</mn></msub><msub><mi>p</mi><mn>2</mn></msub>
<mo>+</mo><mo>⋯</mo><mo>+</mo>
<msub><mi>x</mi><mi>n</mi></msub><msub><mi>p</mi><mi>n</mi></msub>
</mrow>
</math>
E(X) = x_1 p_1 + x_2 p_2 + cdots + x_n p_n
Total[x*p]
EX := add(x[i]*p[i], i = 1..n);
EX = sum(x .* p);
E(X) = x_1 p_1 + x_2 p_2 + … + x_n p_n
Variance (how spread out the values are)
Var(X) = E(X²) − [E(X)]²
\mathrm{Var}(X) = E(X^{2}) - [E(X)]^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>Var</mi><mo>(</mo><mi>X</mi><mo>)</mo>
<mo>=</mo>
<mi>E</mi><mo>(</mo><msup><mi>X</mi><mn>2</mn></msup><mo>)</mo>
<mo>−</mo>
<msup><mrow><mo>[</mo><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>]</mo></mrow><mn>2</mn></msup>
</mrow>
</math>
Var(X) = E(X^2) - [E(X)]^2
Total[x^2*p] - Total[x*p]^2
VX := add(x[i]^2*p[i], i = 1..n) - EX^2;
VX = sum(x.^2 .* p) - EX^2;
Var(X) = E(X^2) - [E(X)]^2
Standard deviation
σ(X) = √Var(X)
\sigma(X) = \sqrt{\mathrm{Var}(X)}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>σ</mi><mo>(</mo><mi>X</mi><mo>)</mo>
<mo>=</mo>
<msqrt><mi>Var</mi><mo>(</mo><mi>X</mi><mo>)</mo></msqrt>
</mrow>
</math>
sigma(X) = sqrt(Var(X))
Sqrt[Total[x^2*p] - Total[x*p]^2]
sigmaX := sqrt(VX);
sigmaX = sqrt(VX);
σ(X) = √(Var(X))
How to have ChatGPT do the calculation
You are a calculation assistant for math (probability and statistics). Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). The probability distribution of a random variable X is given by this table. Value x: 0, 1, 5, 100 Probability p: 0.7, 0.2, 0.09, 0.01 Show each of the following: 1. A check that the probabilities add up to 1 2. The expected value E(X) (both as a fraction in lowest terms or a whole number, and as a decimal) 3. The variance Var(X) (use the formula E(X²) − [E(X)]², and also show the value of E(X²)) 4. The standard deviation σ(X) (as a decimal) In Python, use the fractions module from the standard library to calculate exactly, and show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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