Normal Distribution and Empirical Rule

Summary

This lesson introduces the normal distribution, also known as the Gaussian distribution. It's a symmetric probability distribution with a mean that reflects the data's center. The normal curve's characteristics are defined by its mean and standard deviation, which determine its position and shape. The empirical rule, or 68-95-99.7 rule, is highlighted, stating that nearly all data points fall within three standard deviations of the mean. Specific examples illustrate how to calculate the proportion of data within certain ranges and the proportion of outliers. The script also discusses the binomial distribution's resemblance to the normal distribution, especially when the probability of success is around 0.5 and the number of trials is large.

Takeaways

• 📊 Normal distribution, also known as Gaussian distribution, is a probability distribution that is symmetric about the mean.
• 📈 The normal curve is unimodal and bell-shaped, and any specific normal curve is defined by its mean and standard deviation.
• 🔍 The mean, median, and mode of a normal distribution are all equal and located at the center of the normal curve.
• 📚 Changing the mean moves the normal curve along the horizontal axis without altering its shape.
• 📉 Changing the standard deviation affects the 'flatness' or 'stiffness' of the normal curve; a larger standard deviation results in a flatter curve.
• 📉 The empirical rule, or the 68-95-99.7 rule, states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
• 📐 For a normal distribution with a mean of 150 and a standard deviation of 3, about 68% of data values fall between 147 cm and 153 cm.
• 📈 Approximately 97.5% of heights are greater than 144 cm when the mean height is 150 cm and the standard deviation is 3.
• 📊 About 2.5% of students have a height greater than 156 cm under the same distribution parameters.
• 🔄 The binomial distribution can resemble a normal distribution when the probability of success is around 0.5 and the number of trials is large.

Q & A

What is the normal distribution?

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean. It shows that data near the mean are more frequent than data far from the mean and is represented graphically by a normal curve.

What are the key characteristics of a normal curve?

A normal curve is symmetric about the mean, has a single peak, is unimodal, and is bell-shaped.

How is a normal curve described?

A specific normal curve is described by its mean and standard deviation, which determine its location and flatness.

What happens when you change the mean of a normal curve?

Changing the mean moves the normal curve horizontally along the horizontal axis without changing its shape.

How does the standard deviation affect the normal curve?

The standard deviation affects how flat or steep the normal curve is. A larger standard deviation results in a flatter curve, while a smaller standard deviation results in a steeper curve.

What is the empirical rule or the 68-95-99.7 rule?

The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and 99.7% within three standard deviations.

What proportion of data falls outside of three standard deviations in a normal distribution?

Only about 0.3% of the data falls outside of three standard deviations in a normal distribution.

In the example given, what proportion of grade n students have a height between 147 cm and 153 cm?

Approximately 68% of grade n students have a height between 147 cm and 153 cm, as these values are one standard deviation away from the mean of 150 cm.

What proportion of students have a height greater than 144 cm according to the empirical rule?

97.5% of the students have a height greater than 144 cm, as 2.5% are below one standard deviation from the mean and 95% are within two standard deviations.

How does the proportion of students with a height greater than 156 cm relate to the empirical rule?

Only about 2.5% of students have a height greater than 156 cm, as this value is three standard deviations away from the mean.

When does a binomial distribution resemble a normal distribution?

A binomial distribution resembles a normal distribution when the probability of success is approximately 0.5 and the number of trials (n) is large.

What factors determine if a binomial distribution can be approximated by a normal distribution?

The binomial distribution can be approximated by a normal distribution if the probability of success (p) is close to 0.5 and the number of trials (n) is large enough.