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**"How would you determine if the weather forecast of City A is more reliable than the weather forecast of City B or vice versa?"**

You would *calculate the standard deviation* of City A and City B weather forecast data sets.

But what's standard deviation in the first place?

## Definition Of Standard Deviation

(represented by the symbol sigma, σ) shows how much variation or dispersion (change) exists from the average (mean), or expected value. It is a measure of the average distance between the values of the data in the set and the mean.Standard Deviation

## How Standard Deviation Affects Data Sets?

There's two different kinds of standard deviation: small and large.

### A large standard deviation indicates:

Data points are further away from the mean.

In terms of experimentation, a large standard deviation means that the range of data points is wide and thus, the experiment is harder to replicate and the measurements are less precise.

### A small standard deviation indicates:

Data points are clustered around the mean.

In terms of experimentation, a small standard deviation means that the range of data points is narrow and thus, the experiment is easier to replicate and the measurements are precise.

#### Quiz

A small standard deviation within the data means that there is less variability within the data set

## How To Calculate Standard Deviation?

The following video shows how to calculate standard deviation by using a formula:

### Formula to calculate standard deviation is:

Source: Google Images

If you are extrapolating conclusions from the data set and treating it as a generalizable sample - use n-1

If you are using standard deviation to simply quantify and no extrapolation - use N.

### Steps to calculate standard deviation:

Calculate the mean.

Subtract the mean from each data point.

Square each difference.

Calculate the mean of squared differences.

Take the square root of value obtained in Step 4.

## The Empirical Rule

*The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all observed data will fall within three standard deviations (denoted by σ) of the mean or average (denoted by µ). *

For such data, which occurs only for large number of samples (N>20), the standard deviation has the following meaning:

**68 % of the data would fall within the range of one standard deviation of the average.****95% of the data will fall within the range of two standard deviations.****99% of the data will be within three standard deviations.**

Source: https://i.insider.com/546e68776bb3f74f68b7d0ba?width=750&format=jpeg&auto=webp

#### Quiz

95% of the data will fall within the range of one standard deviation

## Why Study Standard Deviation?

*Product Manufacturing & Quality Control:*

*Product Manufacturing & Quality Control:*

It's essential that car tires that are 2 centimeters in diameters have a small standard deviation to ensure the tire fits appropriately and the car may run without any manufacturing issues! In such a case, measuring the standard deviation is key.

*Finance:*

*Finance:*

Standard deviation is often used as a measure of the risk associated with price fluctuations of a given asset (stocks, bonds, property, etc. ), or the risk of a portfolio of assets.

One of the most important ratios in portfolio management, Sharpe Ratio (for which William Sharpe got a Nobel Prize) uses standard deviation to measure risk-adjusted return (and hence provides incentives to portfolio managers to generate return by taking minimum risk).

*Survey Research:*

*Survey Research:*

If respondents provide data points with high standard deviation, it decreases the reliability of the research.

## Back to.... Weather City Forecast

### "How would you determine if weather forecast of City A is more reliable than weather forecast of City B or vice versa?"

#### Quiz

The temperature is measured in degrees:

## Continued...

### Calculate the mean of both cities' weather forecast

### Calculate the standard deviation of City A weather forecast

### Calculate the standard deviation of City B weather forecast

## The standard deviation of City A weather forecast is 0.89 degree Fahrenheit, whereas the standard deviation of City B is 5.7 degree Fahrenheit. This shows that City A has a more reliable weather forecast than City B because it has a smaller standard deviation.

## Take Action

### During data analysis, make sure to calculate:

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