Standard Deviation Formula

As a former retail chain employee, I understand the challenges of managing excessive data. This one time, I was looking over customer wait times and noticed that the long queues desperately needed a rescue. The standard deviation formula helped me in those days of distress.

I opened my system, entered a few formulas, and voila, suddenly, I knew the numbers even better.

In this tutorial, I’ll cover a range of topics related to equation for standard deviation. Learn from an expert and see how you can ease up data analysis in the flick of a finger.

What is the Standard Deviation Formula?

Now that I've established the solution to all my data problems let me tell you about the one-stop solution standard deviation offers.

The standard deviation formula is a statistical function meant to produce an average and tell you how far each data point falls from the average value in your dataset. But how does it achieve this complexity?

Source: Math is Fun

Formula

σ = √(Σ(x̅ - xi)² / (n - 1))

Let me give you a breakdown of the standard deviation formula Excel often works on:

• Σ (sigma): This one signifies “summation” in plain Greek. It allows you to calculate the total of something.

• (x̅ - xi): The x̅ represents the average value, and xi is the individual data point. Imagine a distance between each point away from the average. When you square this difference (x̅ - xi)², it accounts for both the positive and negative variations.

• N-1: This part refers to the total number of data points (represented by n) minus one.

The standard deviation formula illustrated in steps:

Step 1: Find the difference between each data point and the average value.

Step 2: Square those differences to understand both the positive and negative variations.

Step 3: Add all the squared differences.

Step 4: Divide the sum by n-1. (Here, N accounts for the total data points.)

Formula of Standard Deviation

There are two Excel formulas for use — population standard deviation and sample standard deviation.

1. Population standard deviation formula

You can extract the population standard deviation variance formula by taking the square root of the average of the squared deviations of each value from the population mean (µ), divided by the total population size (N.)

Formula for the population standard deviation

σ = √(Σ(x̅ - xi)² / (n - 1))

where

σ = population standard deviation symbol

x̄ = population mean

xi = terms presented in the data

n = the total number of observations

2. Sample standard deviation formula

The sample standard deviation formula calculates how spread out your data is by finding the square root of the average squared distance from the mean.

Formula for the sample standard formula

s = √(Σ (x̅ - xi)² / n-1)

where

s = sample standard deviation symbol

x̄ = population mean

xi = terms given in the data

n = the total number of observations

Difference between population and sample standard deviation

Why are two types of Deviation Formulae needed?

Among the plethora of reasons, the two most vital are data availability and accuracy.

Let’s say you want to study everyone’s heights in your country. It is highly doubtful that you’ll have access to everyone’s data.

In this case, an equation for standard deviation (sample) may be of better use due to the availability of info on a subset of the population.

When it comes to accuracy, the population standard deviation paints a more accurate picture of the data. On the other hand, think of sample standard deviation as your friend who’s better at estimation.

Standard Deviation Formula for Grouped Data (Discrete Frequency Distribution)

Let’s assume that the exam scores are grouped by grades (A, B, C, D, F), with frequencies representing the number of students in each grade.

Formula using the actual mean method

s = √(Σ f(x - x̅)² / N)

Example

Here are the steps to your standard dev calculator:

Step 1. Calculate the mean (x̅) considering frequencies and grade midpoints (e.g., midpoint of grade A).

Step 2. Find squared deviations (x - x̅)² for each grade.

Step 3. Multiply each squared deviation by its frequency (f).

Step 4. Add the products from step 3 (Σ f(x - x̅)²).

Step 5. Divide the sum by the total students (N) and take the square root.

Sample Standard Deviation Using Three Methods

When working with samples (a portion of the population), I use slightly different formulas. Here’s a quick overview of the three ways in which you can customize your output:

Standard Deviation of Grouped Data by Continuous Method

Alt text: Table showing a coffee shop’s wait time

How to calculate

Step 1. Calculate midpoints (m) for each class except the open-ended ones.

Step 2. Estimate frequency density (f_d) for each class (f / class width). Note that we can't calculate f_d for the open-ended class just yet.

Step 3. Calculate the sample mean (x̅) using midpoints and frequencies.

Step 4. Apply the equation for standard deviation for ungrouped data, using midpoints (m) and frequency densities (f_d) for classes 1-4 only using this formula:

s² ≈ Σ f_d (m - x̅)² / n

Step 5. Take the square root of the result for estimated standard deviation (s).

Step 6. Address the open-ended class. Assume a class width of 20-25 minutes for simplicity.

Step 7. Calculate f_d for this class (assumed to be 2 data points/minute).

Standard Deviation for Probability Distribution

A coin toss experiment is the easiest way to get things going for this method.

The goal is to find the standard deviation (σ) to understand the possible outcomes (heads or tails). The comparison happens with the average outcome (average of getting heads and tails).

How to calculate

Step 1. Calculate the mean (μ): μ = Σ (x * P(x)) = (1 * 0.6) + (0 * 0.4) = 0.6 (average outcome)

Step 2. Calculate squared deviations from the mean: (x - μ)² = [(1 - 0.6)²] + [(0 - 0.6)²] = 0.16 + 0.36

Step 3. Multiply deviations by probabilities: (x - μ)² * P(x) = (0.16 * 0.6) + (0.36 * 0.4) = 0.096 + 0.144

Step 4. Sum the products using: Σ ((x - μ)² * P(x)) = 0.24

By applying the standard deviation formula, here’s your outcome:

σ = √(Σ ((x - μ)² * P(x))) = √(0.24) ≈ 0.49 (rounded)

Standard Deviation for Random Variables

Here’s another example: roll a fair six-sided die. The random variable (X) represents the number rolled (1 to 6).

How to calculate data

Step 1. Define probabilities (each number has a 1/6 chance).

Step 2. Calculate the mean (average outcome, μ = (1 + 2 + ... + 6) / 6 = 3.5).

Step 3. Find squared deviations from the mean for each outcome.

Step 4. Calculate the variance (σ²) considering probabilities (Σ shows summation).

Step 5. Standard deviation (SD) is the square root of the variance (SD = √σ²).

Wrapping Up

Learning the standard deviation formula might appear challenging. However, given the right methods, it can yield accurate (and estimated) results. To learn more about such formulas to ease up your task, visit upGrad.

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Frequently Asked Questions

1. How do I calculate standard deviation?

The method for calculating the standard deviation will depend on whether you’re working with grouped or ungrouped data. To calculate standard division in Excel, use this formula - s = √(Σ (x̅ - xi)² / n-1). In case it is an entire population, use this formula - σ = √(Σ(x̅ - xi)² / (n - 1))

2. What is the standard deviation type formula?

The standard deviation type formula is usually σ = √(Σ(x̅ - xi)² / (n - 1)) for a large group of data and s = √(Σ (x̅ - xi)² / n-1) for sample data.

3. What do you mean by standard deviation?

The Excel standard deviation function allows for calculating the square root of the variant about a dataset.

4. What is Q in the standard deviation formula?

In binomial distribution, the letter ‘Q’ represents the probability of failure. The standard deviation formula for this would be σ = √npq with n standing for the total number of trials, p for the probability of success, and q for the probability of failure.

5. What is the standard deviation of 5 5 9 9 9 10 5 10 10?

For the dataset of 5, 5, 9, 9, 9, 10, 5, 10, 10, the population standard deviation is 2.29.

6. What are 2 standard deviations?

The two types of standard deviations are population standard deviation and sample standard deviation. The former is used for a large dataset, whereas the latter is more useful for a defined subset of data.

7. What is the 5 sigma rule?

The 5 sigma rule is a statistical concept particularly used in hypothesis testing to determine the likelihood of a random chance.

8. What is 1 sigma?

With relation to statistics, 1 sigma (σ) explains one standard deviation from the mean (average) of a distribution. It refers to how spread out the data is in comparison to the average value.

9. What is the 3 sigma limit?

The 3-sigma limit, also known as the 3-sigma rule or the empirical rule, is a statistical concept used primarily for normal distributions (bell-shaped curves). The three determiners are normal distribution, mean, and standard deviation (σ).