Normal Distribution

The normal distribution is one of the most important concepts in statistics and probability. It is a type of probability distribution that is perfectly symmetrical and bell-shaped, where most of the values are concentrated around the mean and fewer values appear as we move away from it. Because of this unique shape, it is often called the bell curve. The normal distribution helps us understand patterns in data, compare results, and make predictions. It is widely used in fields like education (exam scores), healthcare (health data like blood pressure), business (quality control), and finance (stock market analysis).

 

 Table of Contents

• What is Normal Distribution?
• The Normal Distribution Formula
• Understanding the Normal Distribution Curve
• Standard Deviation in Normal Distribution
• Normal Distribution Table (Z-Table)
• Properties of Normal Distribution
• Examples and Solutions Using Normal Distribution
• Applications of Normal Distribution
• Conclusion
• FAQs on Normal Distribution

 

What is Normal Distribution?

The normal distribution is a way to describe how values of a continuous variable are spread out. It is defined using a probability density function, which shows the likelihood of a value occurring within a certain range. For example, if we consider a small interval from $x$ to $x+dx$, the area under the curve in that interval gives the probability that the variable falls there. The graph of a normal distribution is bell-shaped and symmetrical, with most values close to the mean and fewer values far from it. This makes it easier to understand how data behaves and to calculate probabilities for different outcomes.

Here are some key points about the normal distribution:  

• It is a continuous probability distribution.  
• The mean, median, and mode are all the same and are at the center.  
• It is perfectly symmetrical.  
• Data values close to the mean occur more often than values far from it.  

This is why exam scores or students' heights in a class often follow a normal distribution. Most students score near the average, while only a few score very high or very low.

 

The Normal Distribution Formula

The normal distribution formula is the equation used to calculate the probability of a data point in a normal distribution.

Here is the formula:

f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)² / (2σ²))

What do the symbols mean?

• ff(x): The probability density at value xxx

• μ: The mean of the data

• σ: The standard deviation (how spread out the data is)

• π: A constant (3.14159…)

• e: Euler’s number (around 2.718…)

This may look complicated, but you don’t have to memorize it. Just understand that this normal distribution formula helps create the bell-shaped curve.

 

Understanding the Normal Distribution Curve

The normal distribution curve is what we get when we plot the data using the formula. It looks like a smooth, symmetrical bell. Here's how it behaves:

• The center of the curve is the mean.

• The curve is highest at the mean and decreases evenly on both sides.

• The total area under the curve is 1 (or 100%), meaning it represents the entire set of possible values.

• The curve never touches the horizontal line, even far from the mean.

The normal distribution curve is important because it visually shows how data is distributed.

 

Standard Deviation in Normal Distribution

The normal distribution standard deviation tells us how much the values differ from the mean. In a normal distribution:

• About 68% of the values lie within 1 standard deviation of the mean.

• About 95% lie within 2 standard deviations.

• About 99.7% lie within 3 standard deviations.

This is called the Empirical Rule or 68-95-99.7 rule. It’s useful to know how much of the data falls within certain ranges.

Example: If the average height of students is 160 cm and the standard deviation is 10 cm, then:

• 68% of students are between 150 and 170 cm

• 95% are between 140 and 180 cm

• 99.7% are between 130 and 190 cm

Understanding the normal distribution standard deviation helps us know how spread out the data is.

 

Normal Distribution Table (Z-Table)

The normal distribution table, also known as the Z-table, helps us find the probability that a value will fall below or above a certain number in a normal distribution. To use the table, we first convert the value into a Z-score using this formula:  

Z = (X - μ) / σ  

Where:  

• X is the value you are checking,  
• μ is the mean,  
• σ is the standard deviation.  

The Normal Distribution Table (Z score-Table) is also known as a statistical tool that shows the probability values of the standard normal distribution, which is shaped like a bell curve. It helps us find the area under the curve to the left of a given Z-score, where the Z-score tells us how many standard deviations a value lies above or below the mean. Since calculating these probabilities directly involves complex mathematics, the Z-Table provides an easy reference for quickly looking them up. 

Z-Table Values (0.0 to 3.0)

 

 

Properties of Normal Distribution

The properties of the normal distribution make it easy to use in statistics. Here are the main properties:  

• The curve is symmetrical around the mean.  
• It is bell-shaped.  
• The mean, median, and mode are all the same and lie in the center.  
• The curve gets closer to the horizontal axis but never touches it; it is asymptotic.  
• It is defined entirely by its mean (μ) and standard deviation (σ).  
• The total area under the curve is 1, representing 100% probability.  

These properties of the normal distribution help us understand data better and make statistical predictions more precise.  

 

Examples and Solutions Using Normal Distribution

Let’s look at some simple examples to apply what we’ve learned.

Example 1:
Suppose the average score in a class is 70 with a standard deviation of 10. What is the probability that a student scores less than 60?

Solution:
Step 1: Find the Z-score

Z =60 - 70/10

    = −1

Step 2: Look up the value of Z = -1 in the normal distribution table.

The probability is 0.1587.

So, there is a 15.87% chance that a student scores less than 60.

 

Example 2:
What percentage of students scored between 60 and 80?

Step 1: Z for 60 = -1
Z for 80 = +1

From the table:
P(Z < 1) = 0.8413
P(Z < -1) = 0.1587

So, P(60 < X < 80) = 0.8413 - 0.1587 = 0.6826 or 68.26%

 

Example 3:

Q: Find the probability density when x = 3, μ = 4, and σ = 2

Step 1: Normal Distribution Formula
f(x) = 1 / (σ√(2π)) * e^(-(x - μ)² / (2σ²))

Step 2: Substitute values
f(3) = 1 / (2√(2π)) * e^(-(3 - 4)² / (2 * 2²))

Step 3: Simplify inside exponent
(3 - 4)² = 1
2σ² = 8
So, f(3) = 1 / (2√(2π)) * e^(-1/8)

Step 4: Approximate values
√(2π) ≈ 2.506
2√(2π) ≈ 5.012
1 / 5.012 ≈ 0.199
e^(-1/8) ≈ 0.882

Step 5: Multiply
f(3) ≈ 0.199 * 0.882 ≈ 0.176

Final Answer:
f(3) ≈ 0.176

 

Example 4:
Q: Calculate the probability density for x = 5, μ = 5, σ = 1
A:
f(5) = 1 / √(2π) ≈ 0.3989

These examples show how the normal distribution formula, normal distribution table, and standard deviation all come together to solve real-world problems.

 

Applications of Normal Distribution

The normal distribution is common in many areas. Here are some real-life applications of normal distribution:  

• In Education, test scores like the SAT, GRE, or class tests often follow a normal distribution. Most students score around the average, while fewer students score very high or very low.  
• In Business, companies use normal distribution to check product quality. If too many products are far from the average, there may be a production problem.  
• In Healthcare, blood pressure, cholesterol levels, and other health data often fit a normal distribution. Doctors can see if someone's health data is outside the normal range.  
• In Research, scientists use the normal distribution to analyze experiments and draw conclusions.  
• In Stock Markets, investors assume that stock returns are normally distributed to manage risk and predict returns.  

These applications of the normal distribution highlight how important and useful this concept is in everyday life.

 

Conclusion

The normal distribution is one of the cornerstones of statistics because it simplifies the study of data and its spread. Its bell-shaped curve, defined by the mean (μ) and standard deviation (σ), not only explains natural variations but also provides a reliable method for making predictions. By using the normal distribution formula and the Z-Table, we can calculate probabilities, analyze outcomes, and interpret real-world data effectively.

 

Frequently Asked Questions On Normal Distribution

1. What do you mean by normal distribution?

Answer: A normal distribution is a symmetrical, bell-shaped probability distribution where most of the data points cluster around the mean, and probabilities for values taper off equally on both sides. It is used to model many natural phenomena such as height, weight, and test scores.

 

2. What are the 7 properties of normal distribution?

Answer: 

1. It is symmetric about the mean.

2. Mean = Median = Mode.

3. The total area under the curve is 1.

4. The curve is bell-shaped.

5. It follows the empirical rule (68-95-99.7).

6. It is asymptotic to the x-axis.

7. It is completely described by its mean (μ) and standard deviation (σ).

 

3. What is sigma in normal distribution?

Answer: Sigma (σ) represents the standard deviation in a normal distribution. It measures how much the values deviate from the mean. A smaller sigma means data is closely clustered around the mean; a larger sigma indicates more spread.

 

4. What is z called in normal distribution?

Answer: The z in normal distribution is called the z-score. It tells how many standard deviations a data point is from the mean. A z-score of 0 means the value is exactly at the mean.

 

5. What is the normal distribution transformation?

Answer: The normal distribution transformation refers to converting any normal distribution to a standard normal distribution using the formula:
Z = (X - μ) / σ
This standardization allows comparison of different datasets by transforming them to a distribution with mean 0 and standard deviation 1.