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Normal distribution and standard normal distribution are important concepts in statistics. Normal distribution is a bell-shaped distribution that is characterized by its mean and standard deviation, and can be used to describe many natural phenomena. Standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1, and can be used to calculate probabilities of events occurring. Understanding these distributions and how to use them can help us make informed decisions and draw accurate conclusions from data.

The normal distribution is a probability distribution that is widely used in statistics, finance, engineering, social sciences, and natural sciences. It is a continuous distribution that has a bell-shaped curve and is characterized by its mean and standard deviation. The standard normal distribution is a special case of normal distribution in which the mean is 0 and the standard deviation is 1.

In this article, we will explain the characteristics of normal distribution and standard normal distribution, their applications, and how to use them to solve problems.

The normal distribution is a probability distribution that has a bell-shaped curve. It is symmetric around its mean, which is denoted by μ, and the spread of the distribution is determined by its standard deviation, denoted by σ. The probability density function of normal distribution is given by:

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

where x is the random variable, e is the base of natural logarithms, and π is the mathematical constant pi.

The area under the curve of the normal distribution is equal to 1, which means that the total probability of all possible outcomes is 1. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Its probability density function is given by:

```
φ(x) = (1/√(2π)) * e^(-x^2/2)
```

where x is the random variable.

Normal distribution is used in various fields to model the behavior of random variables. Some of its applications are as follows:

- In finance, normal distribution is used to model the behavior of stock prices, interest rates, and other financial variables.
- In engineering, normal distribution is used to model the distribution of measurement errors, the strength of materials, and other engineering variables.
- In social sciences, normal distribution is used to model the distribution of IQ scores, height, weight, and other human characteristics.
- In natural sciences, normal distribution is used to model the distribution of measurements of physical quantities such as temperature, pressure, and mass.
- In statistics, normal distribution is used as a basis for many statistical tests and procedures, including hypothesis testing, confidence intervals, and regression analysis.

The empirical rule, also known as the 68-95-99.7 rule, is used to estimate the percentage of data that falls within a certain number of standard deviations from the mean in a normal distribution. The empirical rule states that:

- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

For example, suppose the distribution of IQ scores is normally distributed with a mean of 100 and a standard deviation of 15. Then, approximately 68% of the population has an IQ score between 85 and 115, approximately 95% of the population has an IQ score between 70 and 130, and approximately 99.7% of the population has an IQ score between 55 and 145.

Z-score is the number of standard deviations from a data point’s mean of a normal distribution. It is calculated by subtracting the mean from the data point and dividing the result by the standard deviation:

```
z = (x - μ) / σ
```

For example, if the height of a population is normally distributed with a mean of 170 cm and a standard deviation of 10 cm, and an individual has a height of 180 cm, the z-score for that individual is (180 - 170) / 10 = 1.

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Z-score can be used to convert any normal distribution to a standard normal distribution, by subtracting the mean and dividing by the standard deviation:

```
z = (x - μ) / σ -> z' = (x - μ) / σ
```

where μ is the mean and σ is the standard deviation of the original distribution, and x is the data point. The new z-score, z', represents the position of the data point relative to the standard normal distribution.

Standard normal distribution can be used to solve problems related to probabilities of events occurring. The area under the curve of the standard normal distribution represents the probability of a random variable taking a value within a certain range. This area can be calculated using tables or using software such as Excel.

For example, suppose the height of a population is normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What is the probability of a randomly chosen person being taller than 190 cm?

We can convert the problem to a standard normal distribution using the formula:

```
z = (x - μ) / σ = (190 - 170) / 10 = 2
```

Using a standard normal distribution table or software, we can find that the probability of a z-score being greater than 2 is approximately 0.0228. Therefore, the probability of a randomly chosen person being taller than 190 cm is 0.0228 or 2.28%.

The central limit theorem states that the sampling distribution of the mean of a large number of independent and identically distributed random variables is approximately normal, regardless of the shape of the underlying distribution. This theorem is important in statistics because it allows us to make inferences about population parameters using sample statistics.

For example, suppose we want to estimate the mean weight of all students in a university. We can take a random sample of students and calculate the mean weight of the sample. The central limit theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the weight distribution of the students in the university.

Normal distribution and standard normal distribution are important concepts in statistics and have many applications in various fields. Understanding these distributions and how to use them can help us make informed decisions and draw accurate conclusions from data.

- Normal distribution is a bell-shaped distribution that is symmetric around the mean and is characterized by its mean and standard deviation.
- The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean.
- Z-score is a measure of how many standard deviations a data point is from the mean of a normal distribution.
- Standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- The area under the curve of the standard normal distribution represents the probability of a random variable taking a value within a certain range.
- The central limit theorem states that the sampling distribution of the mean of a large number of independent and identically distributed random variables is approximately normal, regardless of the shape of the underlying distribution.

**1. What is the formula for calculating z-score?**

a) z = (x - μ) / σ

b) z = (μ - x) / σ

c) z = (x + μ) / σ

d) z = (x - μ) * σ

**Answer**: a) z = (x - μ) / σ

**2. The central limit theorem states that:**

a) the mean of a normal distribution is zero

b) the standard deviation of a normal distribution is one

c) the sampling distribution of the mean of a large number of independent and identically distributed random variables is approximately normal

d) the sampling distribution of the mean of a small number of independent and identically distributed random variables is approximately normal

**Answer**: c) the sampling distribution of the mean of a large number of independent and identically distributed random variables is approximately normal

**3. What is the standard deviation of a standard normal distribution?**

a) 0

b) 1

c) 2

d) It depends on the mean.

**Answer**: b) 1

**4. The empirical rule states that:**

a) approximately 68% of the data falls within one standard deviation of the mean

b) approximately 95% of the data falls within two standard deviations of the mean

c) approximately 99.7% of the data falls within three standard deviations of the mean

d) all of the above

**Answer**: d) all of the above

Module 5: Probability Distributions

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