Question

What proportion of a normal distribution is within one standard deviation of the mean? (b) What proportion is more than 2.0 standard deviations from the mean? (c) What proportion is between 1.25 and 2.1 standard deviations above the mean?

Short answer

a) 68%, b) 5%, c) 8.77%.

Step-by-step solution

Understanding the Normal Distribution

A normal distribution is symmetric around its mean. It's defined by its mean (\(\mu\)) and standard deviation (\(\sigma\)), and is often visualized as a bell curve.

Proportion Within One Standard Deviation

According to the empirical rule, about 68% of the data in a normal distribution falls within one standard deviation of the mean, i.e., between \(\mu - \sigma\) and \(\mu + \sigma\).

Proportion More Than Two Standard Deviations

The empirical rule also states that about 95% of the data falls within two standard deviations of the mean. Therefore, the proportion more than two standard deviations from the mean is \(1 - 0.95 = 0.05\), or 5%, but we must consider both tails. Thus, half this proportion (2.5%) is beyond ±2 standard deviations.

Proportion Between 1.25 and 2.1 Standard Deviations

Using standard normal distribution tables or a calculator: Find \(P(Z < 2.1)\) and \(P(Z < 1.25)\) where \(Z\) is the standard score. - \(P(Z < 2.1)\) is approximately 0.9821.- \(P(Z < 1.25)\) is approximately 0.8944.The proportion between these values is \(0.9821 - 0.8944 = 0.0877\), or 8.77%.

Key concepts

Standard Deviation

In statistics, the standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation means that the data points are generally close to the mean of the dataset, while a high standard deviation indicates that the data points are spread out over a wider range of values.

When we talk about normal distribution, the standard deviation is crucial because it determines the width of the bell curve. Specifically, it tells us how much observations tend to deviate from the mean in a normal distribution.

Key points about standard deviation in a normal distribution:

• The mean (average) value lies at the center of the bell curve.
• Most data falls within one standard deviation of the mean in a normal distribution.
• It is represented by the Greek letter \( \sigma \) (sigma).

Understanding the standard deviation helps to identify how typical or unusual a data point is from the average or expected value.

Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule stating that for a normal distribution:

• Approximately 68% of the data falls within one standard deviation (\( 1\sigma \)) of the mean.
• About 95% of the data falls within two standard deviations (\( 2\sigma \)) of the mean.
• Nearly 99.7% of the data falls within three standard deviations (\( 3\sigma \)) of the mean.

This rule is incredibly useful for quickly estimating the spread of data in a normal distribution, which is symmetric and bell-shaped. By understanding the empirical rule, students can easily determine the proportion of data within certain ranges of the mean.

For example:

• If you're asked what proportion of data lies within one standard deviation from the mean, the empirical rule quickly tells us it's around 68%.
• The proportion more than two standard deviations away is what remains outside 95%, or 5% in total, split across both tails of the distribution.

This rule simplifies complex calculations, offering a quick glimpse into the behavior of data in a normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution that has a mean of 0 and a standard deviation of 1. It is often represented using the letter \( Z \), hence the term "Z-score" comes into play. A Z-score measures how many standard deviations an element is from the mean.

Properties of the standard normal distribution include:

• The distribution curve is symmetric about the mean, which is 0.
• The total area under the curve is 1.
• Z-scores enable comparison of data points from different normal distributions.

To find the proportion of the normal distribution that lies between two Z-scores, you can consult a Z-table or use statistical calculators.

For instance, if we want to know the proportion between Z-scores of 1.25 and 2.1, we determine:

• The cumulative probability for Z=2.1 is approximately 0.9821.
• The cumulative probability for Z=1.25 is roughly 0.8944.
• The proportion between these Z-scores is the difference, which is 8.77%.

This clarity helps in understanding where data points lie relative to the mean in terms of standard deviations, simplifying complex probability calculations.