Understanding Mean and Standard Deviation in Normally Distributed Data

In statistics and data analysis, the mean and standard deviation are crucial metrics that help us understand the distribution of data points. When the data is normally distributed and spans a given range, such as between 51 and 64, we can deduce the mean and standard deviation using the properties of the normal distribution.

The Importance of the Normal Distribution

Normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics. It is characterized by its bell-shaped curve and symmetrical nature. In a normal distribution:

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

These properties allow us to make probabilistic statements about the data based on its distribution.

Problem Description

Let's consider a normally distributed dataset where 68 of the data falls between two specific values, 51 and 64. We aim to find the mean (μ) and the standard deviation (σ) of this dataset.

Step-by-Step Solution

Step 1: Setting Up the Equations

Given that in a normal distribution, approximately 68% of the data lies within one standard deviation (σ) from the mean (μ), we can set up the following system of equations:

Lower Bound (51): μ - σ Upper Bound (64): μ σ

These equations can be written as:

μ - σ 51

μ σ 64

Solving for the Mean (μ)

Add the two equations to solve for μ:

(μ - σ) (μ σ) 51 64

2μ 115

μ 115 / 2 57.5

Solving for the Standard Deviation (σ)

Subtract the first equation from the second to solve for σ:

(μ σ) - (μ - σ) 64 - 51

2σ 13

σ 13 / 2 6.5

Therefore, the mean and standard deviation of the dataset are:

Mean (μ): 57.5 Standard Deviation (σ): 6.5

Conclusion

By utilizing the properties of the normal distribution and the given range of data, we can accurately determine the mean and standard deviation. In this case, the mean is 57.5 and the standard deviation is 6.5, providing a clear understanding of the dataset's central tendency and dispersion.

Further Insights and Applications

Understanding mean and standard deviation is essential in various fields, including finance, biology, and engineering. It allows us to make informed decisions and draw meaningful conclusions from data. For instance, in finance, normal distribution and these metrics help in risk assessment and portfolio management.

By mastering these concepts, analysts can effectively model and predict data behavior, making the data analysis process more robust and reliable.

Keywords

Mean Standard Deviation Normal Distribution