How to Find Standard Deviation of Probability Distribution (original) (raw)

Last Updated : 23 Jul, 2025

**Standard Deviation is a measure in statistics that determines the amount of variability or dispersion in a set of values. Understanding how to calculate the standard deviation is useful for analyzing the variability or spread of random variables in probability distributions.

In this article, we will learn about standard deviation and how to find the standard deviation of a probability distribution.

Table of Content

What is Probability Distribution?

The probability distribution describes how probabilities are distributed over the values of a random variable. It outlines all possible outcomes of a random event and the likelihood of each outcome occurring.

For a discrete random variable, it lists each value and its associated probability. For a continuous random variable, it is described by a probability density function (PDF).

Types of Probability Distributions

What is Standard Deviation?

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much individual data points differ from the mean (average) value of the dataset. It is denoted by the symbol σ (sigma).

Standard Deviation Definition

Standard deviation (often denoted as σ for a population or s for a sample) measures the average amount of variability or dispersion of a dataset around its mean.

What is Standard Deviation Of Probability Distribution?

The standard deviation of a probability distribution measures the spread or dispersion of the possible values of a random variable around the mean of the distribution. It provides a numerical measure of how much the values deviate from the expected value (mean) of the distribution.

In simpler terms, it indicates how much the potential results deviate from the average result that one expect based on the probabilities assigned to each outcome.

How to Find Standard Deviation of Probability Distribution

Below is a step-by-step process to find the standard deviation of a probability distribution:

**Step 1: Calculate the Expected Value (Mean): For each possible outcome (x) in your distribution, multiply it by its probability (P(x)) and then sum these products (Σ(x × P(x))). This sum represents the expected value (μ) of the distribution.

**Step 2: Calculate Deviations from the Mean: For each outcome (x), subtract the expected value (μ) from it (x - μ). This represents the deviation of that outcome from the average.

**Step 3: Square the Deviations: Square the deviations you calculated in step 2 [(x - μ)²].

**Step 4: Weight Squared Deviations by Probability: Multiply the squared deviation [(x - μ)²) of each outcome by its probability (P(x)).

**Step 5: Sum Weighted Deviations: Sum the products you obtained in step 4 (Σ((x - μ)² × P(x))). This sum represents the variance of the distribution, which is essentially the average squared deviation from the mean.

**Calculate the Standard Deviation: Take the square root of the variance you calculated in step 5. σ = √(Σ((x - μ)² × P(x)))

Applications of Standard Deviation

Standard deviation is an important statistical measure that quantifies the amount of variation or dispersion of a set of values. It finds applications in numerous fields. Few of them are added below:

In Finance

In Quality Control

In Weather Forecasting

In Health and Medicine

In Market Research

Examples on Standard Deviation of Probability Distribution

**Example 1: A bag contains 5 red marbles, 3 blue marbles, and 2 yellow marbles. You pick one marble without looking. What is the standard deviation of the color (red, blue, yellow)? (Consider assigning 1 for red, 2 for blue, and 3 for yellow).

**Solution:

**Calculate the mean first

The mean μ of the random variable X is calculated using the probabilities of each color:

μ = E(X) = 1⋅P(Red) + 2⋅P(Blue) + 3⋅P(Yellow)

Calculate P(Red), P(Blue), and P(Yellow):

P(Red) = 5/10 = 0.5

P(Blue) = 3/10 = 0.3

P(Yellow) = 2/10 = 0.2

μ = 1⋅0.5 + 2⋅0.3 + 3⋅0.2 = 0.5 + 0.6 + 0.6 = 1.7

**Calculate the Variance, σ 2 :

σ2 = Var(X) = [(1 - 1.7)2 × 1/2] + [(2 - 1.7)2 × 3/10] + [(3 - 1.7)2 × 1/5] = 0.49

Standard Deviation = √Var(X) = √0.49 = 0.7

**Example 2: A fair coin is flipped 10 times. Find the standard deviation of the number of heads.

**Solution:

This is a binomial distribution with n = 10 and p = 0.5.

The standard deviation of a binomial distribution is given by: σ = √(npq)

σ = √(10 × 0.5 × 0.5) = √2.5 ≈ 1.58

Therefore, the standard deviation is approximately 1.58.

**Example 3: A grocery store on average receives 5 customers every 15 minutes. Find the standard deviation for the number of customers arriving in a 30-minute interval.

**Solution:

**Step 1: Find the average rate for 30 minutes

Since the average rate for 15 minutes is 5 customers, the average rate for 30 minutes would be 5 × 2 = 10 customers.

**Step 2: Use the Poisson distribution property

For a Poisson distribution, the mean and variance are equal. Therefore, the variance for the number of customers in 30 minutes is also 10.

**Step 3: Calculate the standard deviation

Standard deviation is the square root of the variance.

Standard deviation = √10 ≈ 3.16

Therefore, the standard deviation for the number of customers arriving in a 30-minute interval is approximately 3.16.

**Example 4: A random variable X is uniformly distributed over the interval [2, 5]. Find the standard deviation.

**Solution:

For a uniform distribution over the interval [a, b], the standard deviation is given by: σ = (b - a) / √12

σ = (5 - 2) / √12 = 3 / √12 ≈ 0.87

**Example 5: The average number of accidents per day on a highway is 3. Find the standard deviation of the number of accidents per day.

**Solution:

This is a Poisson distribution with λ = 3.

Standard deviation of a Poisson distribution is the square root of the mean, σ = √λ.

σ = √3 ≈ 1.73

**Example 6: An urn contains 4 red balls and 6 blue balls. You draw two balls without replacement. What is the standard deviation of the sum of the balls' colors (considering 1 for red and 2 for blue)?

**Solution:

Determine Possible Outcomes and Their Probabilities:

RR: Probability = (4/10) × (3/9) = 2/15

RB or BR: Probability = 2 × (4/10) × (6/9) = 8/15

BB: Probability = (6/10) × (5/9) = 1/3

Calculate the Expected Value (Mean):

E(X) = (2 × 2/15) + (3 × 8/15) + (4 × 1/3) = 2.8

Calculate the Variance:

Var(X) = [(2 - 2.8)2 × 2/15] + [(3 - 2.8)2 × 8/15] + [(4 - 2.8)2 × 1/3] = 0.32

Calculate the Standard Deviation:

SD(X) = √Var(X) = √0.32 ≈ 0.5657

Practice Questions on Standard Deviation of probability distribution

**Q1: A fair coin is tossed. What is the standard deviation of the probability distribution of getting heads (H) or tails (T)?

**Q2: What is the standard deviation of the probability distribution when rolling a fair six-sided die?

**Q3: You draw one card from a standard deck of cards without replacement. What is the standard deviation of the assigned point value (Ace = 1, Jack/Queen/King = 10, others = face value)?

**Q4: A fair coin is flipped 5 times. What is the standard deviation of the probability distribution of getting exactly 2 heads?

**Q5: A customer service center receives an average of 3 calls per hour. What is the standard deviation of the probability distribution of the number of calls received in a 2-hour period?

**Q6: A machine dispenses candy with 4 different colors (red, green, blue, yellow) with equal probability. What is the standard deviation of the probability distribution of getting a specific color candy?

**Q7: The time it takes a light bulb to burn out follows an exponential distribution with an average lifespan of 1000 hours. What is the standard deviation of the probability distribution of the bulb's lifespan?

**Q8: The heights of adults in a town are normally distributed with an average height of 170 cm and a standard deviation of 5 cm. What is the standard deviation of the probability distribution of heights?

**Q9: You draw two cards from a standard deck of cards without replacement. What is the standard deviation of the sum of the assigned point values?

**Q10: Imagine you run a bakery that sells cupcakes. You know the average number of cupcakes sold daily is 20 with a standard deviation of 3 cupcakes. How can the standard deviation help you with your business decisions (e.g., inventory management)?

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