**What is Normal Distribution?**

**The normal distribution**, often depicted as a bell-shaped curve, is a fundamental concept in statistics.

It describes how data points are spread around the mean, with most data clustering near the average and fewer observations in the tails.

**Usages in Business**

**Human Resources**: In performance evaluations, a majority of employees might perform at an average level, with fewer excelling or underperforming.**Sales**: Sales data for a popular product might follow a normal distribution, indicating consistent sales patterns.**Finance**: Stock market returns can often be modeled using normal distribution, aiding in risk assessment.

**Calculating with Normal Distribution**

The formula for a normal distribution is:

- represents the probability density function.
- is the mean or average.
- stands for the standard deviation.
- is the base of the natural logarithm.

You might be overwhelmed to see this formula, but **you don’t have to remember it **in most cases.

**Empirical Rule**

**The Empirical Rule** is a fundamental aspect of the normal distribution.

It provides a quick estimate of the probability of a value occurring within certain ranges:

**68.2%**of data falls within**one standard deviation (σ)**of the mean (µ).**95.4%**within**two standard deviations (2σ)**.**99.7%**within**three standard deviations (3σ)**.

**Skewness**

Skewness measures the degree of asymmetry of a distribution.

**A perfectly normal distribution is symmetric, meaning its skewness is zero.** If the distribution leans to the left or right, it’s said to be skewed.

**Positive Skewness (right-skewness)**: The right tail is longer than the left. This means there are exceptionally high values.**Negative Skewness (left-skewness)**: The left tail is longer, indicating exceptionally low values.

**Kurtosis**

Kurtosis gauges the thickness of the tail ends of a distribution compared to a normal distribution. The normal distribution has a kurtosis of 3.0. Distributions can be:

**Leptokurtic (>3.0 kurtosis)**: They have “fat tails” or more extreme values than a normal distribution.**Platykurtic (<3.0 kurtosis)**: They have “skinny tails” or fewer extreme values.

**Using Normal Distribution**

Let’s use an example to illustrate **the application of the normal distribution**.

Suppose in a school, the average height of students **(µ) is 170cm**, and the standard deviation **(σ) is 5cm**.

Based on that µ and σ, we can say the following things.

**1. 165cm (µ – σ) **

A student with a height of **165cm** (µ – σ) would be taller than approximately **16%** of the students (since 68% is between µ – σ and µ + σ, and it’s symmetric).

**2. 180cm (µ + 2σ)**

A student with a height of **180cm** (µ + 2σ) would be taller than about **97.7%** ( 2.3%+95.4%) of the students.

**3. Probability: Student ≥185cm**

The probability of finding a student with a height of** 185cm** (µ + 3σ) is approximately 0.15%, because 99.7% of students falls within 185cm ~ 150cm according to the empirical rule.

To calculate these probabilities, we use the **Z-score** formula and then look up the value in a **Z-table** or use statistical software.

See Also: Z-Score in Statistics: Definition, Formula with Examples

**Types of Distribution**

The world of statistics is vast, and while the normal distribution is a cornerstone, **there are several other distributions** that play crucial roles in various scenarios.

Let’s explore some of these distributions, their formulas, and understand how they differ from the normal distribution, especially focusing on the differences in their formulas.

**Normal Distribution**

**Formula**

Remember, is the mean and is the standard deviation.

This distribution is symmetric and bell-shaped.

**Uniform Distribution**

When following the Uniform Distribution, every outcome is equally likely within a given range.

**Formula**

**Difference from Normal Distribution**

The uniform distribution is flat, meaning every outcome in the range [a, b] has an equal probability.

Unlike the normal distribution, which has a peak at the mean, the uniform distribution doesn’t favor any particular value.

**Binomial Distribution**

Binomial Distribution represents **the number of successes** in a fixed number of Bernoulli trials.

**Formula**

**Difference from Normal Distribution**

The binomial is discrete and depends on (number of trials) and (probability of success).

The shape can vary based on . If is large and is close to 0.5, it can look similar to a normal distribution, but it’s fundamentally about discrete success/failure scenarios.

**Poisson Distribution**

Poisson Distribution represents the number of events in fixed intervals of time or space.

**Formula**

**Difference from Normal Distribution**

The Poisson focuses on the number of events in a fixed interval and is defined by (average rate of value). It’s used for rare events and can approach the shape of a normal distribution when is large.

**Exponential Distribution**

Exponenial Distribution describes the time between events in a Poisson process.

**Formula**

**Difference from Normal Distribution**

The exponential distribution is **continuous but skewed, with a peak at zero**. It’s defined by , the rate parameter.

Unlike the symmetric bell curve of the normal distribution, the exponential has a long tail.

By understanding the differences in these formulas and their implications, one can better grasp the unique characteristics and applications of each distribution in contrast to the normal distribution.