**What is Standard Deviation?**

Standard Deviation (often represented as σ) is a measure that tells us how spread out the numbers in a data set are around the mean (average).

In simpler terms, it shows **how much individual data points differ from the average**.

The higher the standard deviation, the more spread out the data points are; the lower it is, the closer the data points are to the mean.

*Formula:*

- = Standard Deviation
- = Each individual data point
- = Mean of the data set
- = Number of data points

**How to Calculate Standard Deviation**

Standard Deviation can be calculated by following these steps:

**Find the Mean****Calculate the Difference from the Mean****Square the Differences****Find the Average of the Squared Differences****Take the Square Root**

**Excercise: Calculate Standard Deviation**

Let’s take a simple dataset as an example:

**[ 4, 8, 6, 5, 3, 2, 8, 9, 7 ]**

**1. Find the Mean**

To find the mean, sum up all the numbers and then divide by the number of data points.

(rounded to two decimal places)

The mean gives us a central value for our dataset, which will be used to determine **how far each data point deviates from this center**.

**2. Calculate the Difference from the Mean**

Subtract the mean from each data point to see how much each point deviates from the mean.

This step helps us understand how each data point varies from the average. It sets the stage for the next steps where we quantify this variation.

**3. Square the Differences**

Squaring serves two main purposes:

- It eliminates
**negative differences**, ensuring all values are positive. - It gives more weight to larger deviations, emphasizing points that are further from the mean.

**4. Find the Average of the Squared Differences (Variance)**

Sum up all the squared differences and divide by the number of data points.

(rounded to two decimal places)

This step gives us **the average squared deviation from the mean**, which is a measure of the spread or dispersion of our data.

**5. Take the Square Root (Standard Deviation)**

(rounded to two decimal places)

Taking the square root brings our measure of spread (which was squared in the variance) **back to the original unit of measurement**.

The result, the standard deviation, tells us, on average, how far each data point is from the mean.

**6. Done**

The standard deviation, in this case, is **1.61**.

**This means that most data points are, on average, 1.61 units away from the mean of our dataset**.

It’s a crucial measure in statistics, providing insight into the dispersion or spread of our data.

**Standard Deviation vs Variance**

While both standard deviation and variance measure the spread of data points, they are used in slightly different contexts.

The key difference is that variance gives the average of the squared differences from the Mean, while standard deviation is **the square root** of this variance.

In essence, **standard deviation is a more interpretable metric** as it’s in the same unit as the data, whereas variance is in squared units.

**Types of Standard Deviation**

Standard deviation can be classified into two main types based on the dataset being analyzed: Population Standard Deviation and Sample Standard Deviation.

Let’s delve deeper into each type:

**1. Population Standard Deviation**

This is used when considering **the full set of data**, meaning every member of the population is considered.

It provides a precise measure of spread for the entire dataset.

**Key Point**

Since it considers the entire population, it gives a definitive measure of variability.

**Formula**

- = Population Standard Deviation
- = Each individual data point
- = Mean of the entire population
- = Total number of data points in the population

**2. Sample Standard Deviation**

This is used when analyzing a subset or sample from a larger population.

It provides an estimate of the spread based on the sample, which might be used to infer the spread of the entire population.

**Key Point**

The formula for the sample standard deviation divides by (instead of ( N ) as in the population standard deviation).

This adjustment, known as Bessel’s correction, is made to give a more unbiased estimate of the population standard deviation.

**Formula**

- = Sample Standard Deviation
- = Each individual data point in the sample
- = Mean of the sample
- = Number of data points in the sample

**Population SD vs Sample SD**

The primary distinction between the two lies in the dataset they analyze and the divisor used in their formulas.

Population standard deviation considers every data point in the entire population and divides by , while the sample standard deviation considers **only a subset of the population** and divides by .

The reason for the divisor in the sample standard deviation is to provide **a more accurate and unbiased estimate** of the population’s standard deviation, especially when the sample size is small.

**Usages of Standard Deviation in Business**

**1. Usage in Finance**

Investors use standard deviation to measure the volatility of stocks or investment portfolios.

A higher standard deviation indicates higher risk but potentially higher returns.

**2. Marketing**

In market research, a high standard deviation might indicate a wide range of opinions or behaviors among a target audience, suggesting the need for segmented marketing strategies.

**3. Quality Control**

Manufacturers use standard deviation to ensure consistent product quality. A low standard deviation indicates consistent manufacturing processes.