Investment Guide
# What is Standard Deviation?

**How does Standard Deviation Work?**

**Standard Deviation Formula**

#### For a population:

#### For a sample:

**Understanding Standard Deviation**

**Comparing Standard Deviation**

**Importance of Standard Deviation**

**Finance – Portfolio Management and Investment**

**Banking – Credit Scoring**

**Conclusion**

Standard deviation is a statistical measure that tells us how spread out the values in a dataset are around the mean (average). It is a crucial element of statistical analysis, offering insights into variability and helping to make sense of the data at hand. In this article, we will examine what standard deviation is, how it works, its formula and why it is important.

Standard deviation measures the degree to which individual data points in a dataset vary or 'deviate' from the mean, or average, of the data set. In other words, it gives a sense of how spread out the values in a dataset are.

If the data points are generally close to the mean, the standard deviation is small. If data points are widely spread out from the mean, the standard deviation is large. For instance, if we have two data sets with the same mean, let us say 100, the one with values more concentrated around 100 will have a smaller standard deviation than the one with values scattered more widely.

The process of calculating the standard deviation involves a few steps. The formula for calculating the standard deviation depends on whether you have data for an entire population or a sample.

σ = sqrt[ Σ ( Xi - μ )^2 / N ]

Where:

- σ is the population standard deviation,
- Xi represents each value from the population,
- μ is the population mean,
- N is the total number of values in the population.

s = sqrt[ Σ ( xi - x̄ )^2 / (n - 1) ]

Where:

- s is the sample standard deviation,
- xi represents each value from the sample,
- x̄ is the sample mean,
- n is the total number of values in the sample.

Understanding standard deviation involves recognising its role in illustrating the dispersion within a dataset. Let us consider an example.

Suppose you have two classes of students and their respective scores on a test.

Class A: 85, 90, 95, 100, 105; Class B: 70, 80, 95, 110, 120

Both classes have an average (mean) score of 95, but when we calculate the standard deviation for each class, we get about 7.9 for Class A and 20.6 for Class B.

This tells us that even though the average score is the same for both classes, the scores in Class B are more spread out from the mean compared to Class A.

Comparing standard deviations is a way to understand variability across different datasets or within subsets of a dataset.

If you are comparing two datasets with different means, you might find that one has a higher standard deviation than the other. This does not necessarily mean that one dataset is more variable than the other, as the standard deviation is influenced by the mean. A dataset with a larger mean might have a larger standard deviation simply because the values themselves are larger, not because they are more spread out.

When comparing standard deviations, it is crucial to consider the context. Comparing the standard deviations of datasets with similar means can give a good sense of which dataset is more variable. Similarly, comparing the standard deviations of subsets within a dataset can reveal whether some subsets are more variable than others.

Standard deviation is important in several ways.

First, it is a fundamental measure of variability. Whether you are a researcher analysing data from an experiment, a businessperson looking at sales trends or a policymaker examining population data, knowing the standard deviation can help you understand the range of possible outcomes and how much they are likely to deviate from the average.

Moreover, the standard deviation is a crucial component of many statistical tests and procedures. It is used in calculating confidence intervals, conducting hypothesis tests and creating statistical models.

In finance, particularly in portfolio management and investment, the standard deviation measures the volatility or market risk associated with an investment. Higher standard deviation values indicate higher volatility and therefore higher risk, while lower values suggest lower risk.

For instance, let us consider two investment funds, Fund A and Fund B, with the same average annual return of 8%. However, Fund A has a standard deviation of 4%, while Fund B's standard deviation is 10%. This suggests that returns for Fund A are more predictable and less volatile than Fund B. Therefore, conservative investors, who prioritise steady returns over high gains, might prefer Fund A due to its lower risk.

In banking, the standard deviation is used in the credit scoring process, which predicts the probability of a borrower defaulting on a loan. Financial institutions gather various data points, such as the borrower's income, existing debts and credit history.

Consider that a bank has data on borrowers' incomes with a mean of Rs. 50,000 and a standard deviation of Rs. 10,000. If a particular borrower's income is Rs. 70,000, we know that their income is two standard deviations above the mean. This information, along with similar computations on other data points, feeds into a model that generates a credit score.

In this case, the standard deviation helps the bank understand how typical or atypical each piece of data is. Borrowers with data that is closer to the mean on all data points might be seen as less risky, given that they represent the 'typical' borrower.

These examples underline the value of the standard deviation in real-world financial and banking situations. By quantifying risk and variability, it allows finance professionals and banks to make better-informed decisions.

The standard deviation allows for a clear comparison between different datasets. By comparing standard deviations, we can infer the relative volatility or dispersion in different sets of data, which can guide decision-making and risk assessment.

A standard deviation is a powerful tool in statistics. By providing a measure of variability, it offers a deeper understanding of the dataset, enabling more informed decision-making and risk assessment. Its importance cannot be understated in fields that depend heavily on data, such as finance, economics, psychology and many more.

Tata Capital, like any other financial institution, makes extensive use of standard deviation in its various operations. In their loan and credit operations, standard deviation helps analyse potential borrowers' data to predict creditworthiness and manage risk.

Thus, standard deviation plays a vital role in financial institutions such as Tata Capital, where it helps assess risk, guide decision-making and ultimately ensure financial stability. It is a testament to the versatility and utility of standard deviation as a statistical tool.