Normal Distribution

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Normal Distribution

By Rasel Barte

LEARNING OBJECTIVES ​

• Explain normal distribution;

• Appreciate the importance of normal distribution in real - life through oral participation; and

• Solve the z - scores of standard normal distribution. ​

Random variables​

​A random variable, x, is defined as a variable whose values are determined by chance, such as the outcome of rolling a die.

​A continuous variable is a variable that can assume any values in an interval between any two given values.

​For example, height is a continuous variable. A person’s height may theoretically be any number greater than zero.

Multiple Select

What are other example of continuous variables?

• Height of the students

• number of patients in the hospital

• number of petals in a flower

• length of your hair

The histogram shows the heights of a sample of American women.

The histogram is a symmetrical bell-shape. Distributions with this shape are called normal distributions.​

​A curve drawn through the top of the bars approximates a normal curve.

​In an ideal normal distribution the ends would continue infinitely in either direction. However, in most real distributions there is an upper and lower limit to the data.

​When the mean is 0 and the standard deviation is 1, this is called the standard normal distribution.

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​Some text here about the topic of discussion.

​A normal distribution is defined by its mean and variance. These are parameters of the distribution.

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The normal distribution:  x ~ N(μ, σ²)

The random variable, x, has a normal distribution of mean, μ, and variance, σ².​

The normal distribution is a continuous distribution.

In a continuous distribution, the probability that a random variable will assume a particular value is zero. Explain why.​

​For a discrete random variable, as the number of possible outcomes increases, the probability of the random variable being one particular outcome decreases.

A continuous variable may take on infinitely many values, so the probability of each particular value is zero.

​This means that the probability of the random variable falling within a range of values must be calculated, instead of the probability of it being one particular value.

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​Since all probabilities must fall between 0 and 1 inclusive, the area under the normal distribution curve represents the entire

sample space, thus it is equivalent to 100% or 1.

​The probability that a random variable will lie between any two values in the distribution is equal to the area under the curve between those two values.

​Area under the curve

The mean divides the data in half. If the area under the curve is 1.00 then the area to one side of the mean is:

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1 × 0.5 = 0.5

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​The empirical rule

The standard deviation and the mean together can tell you where most of the values in your distribution lie if they follow a normal distribution.

The empirical rule, or the 68-95-99.7 rule, tells you where your values lie:

• Around 68% of scores are within 1 standard deviation of the mean,

• Around 95% of scores are within 2 standard deviations of the mean,

• Around 99.7% of scores are within 3 standard deviations of the mean.