How to Find Standard Deviation of Probability Distribution
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.
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
- Discrete Probability Distribution: Deals with finite or countable outcomes. Example: Number of heads in 3 coin flips.
- Continuous Probability Distribution: Deals with infinite and uncountable outcomes. Example: Height of people.
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
- Risk assessment: Standard deviation is used to measure the volatility of investment returns. A higher standard deviation indicates higher risk.
- Portfolio management: Investors use it to diversify portfolios and manage risk.
In Quality Control
- Process control: Standard deviation helps monitor process variability and identify when a process is out of control.
- Product consistency: It ensures product quality by measuring deviations from target specifications.
In Weather Forecasting
- Predictability: Standard deviation helps in understanding the variability of weather patterns.
- Risk assessment: It is used to assess the potential impact of extreme weather events.
In Health and Medicine
- Clinical trials: Standard deviation is used to measure the variability in treatment responses.
- Epidemiology: It helps analyze the distribution of diseases and risk factors.
- Medical research: It is essential for statistical analysis of medical data.
In Market Research
- Customer segmentation: Standard deviation helps identify different customer groups based on purchasing behavior.
- Market analysis: It is used to understand market trends and fluctuations.
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, σ2:
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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