Understanding Mean and Standard Deviation in a Normal Distribution Given Precision

In the context of normal distribution, approximately 68.3% of the values fall within one standard deviation of the mean. This property is fundamental for a wide range of data analysis and statistical applications. Given that 68.3% of the values in a specific dataset fall between 2.8 and 4.0, we can determine the mean and standard deviation of the distribution. Let's explore how to do it.

Given Condition and Set Up Equations

Given that 68.3% of the values lie between 2.8 and 4.0, we set up the following equations using the mean ( mu ) and standard deviation ( sigma ):

Lower bound: [ mu - sigma 2.8 ]

Upper bound: [ mu sigma 4.0 ]

Solving the Equations

Step 1: Add the two equations.

[ (mu - sigma) (mu sigma) 2.8 4.0 ] [ 2mu 6.8 ] [ mu frac{6.8}{2} 3.4 ]

Step 2: Substitute ( mu 3.4 ) back into one of the original equations to find ( sigma ).

Substituting into the lower bound equation:

[ 3.4 - sigma 2.8 ] [ sigma 3.4 - 2.8 0.6 ]

Thus, the mean ( mu ) and standard deviation ( sigma ) of the distribution are:

Mean ( mu ): 3.4

Standard Deviation ( sigma ): 0.6

Verification and Additional Insights

The property of the Normal distribution also tells us that approximately 68% of the area under the curve lies between plus and minus one standard deviation from the mean. For a data set that follows this distribution, the standard deviation can be derived from the given range. Let's verify our solution using a different approach:

Alternate Method

We can calculate the standard deviation by knowing that 68.3% of the values lie between 2.8 and 4.0.

1. Since 68.3% of the values fall between two standard deviations around the mean, we can use the formula:

P(2.8 X 4.0) P(0.6 Z 1.2)

From the standard normal distribution table, we know that:

P(Z 1.2) 0.8849

Thus:

P(0.6 Z 1.2) 0.8849 - 0.5 0.3849 0.2 0.683

Here, ( Z frac{4.0 - mu}{sigma} ) and ( Z frac{2.8 - mu}{sigma} ).

Given ( Z 1.2 ) and solving for ( sigma ):

[ 1.2 frac{4.0 - 3.4}{sigma} ] [ 1.2 frac{0.6}{sigma} ] [ sigma frac{0.6}{1.2} 0.5 ]

Alternatively, using the left side of the interval:

[ -0.6/sigma -1 ] [ sigma frac{0.6}{1} 0.6 ]

Therefore, the mean ( mu ) and standard deviation ( sigma ) are consistent across different approaches.

Conclusion

By applying the properties of the normal distribution and solving the given data, we can confidently say that the mean and standard deviation of the distribution are 3.4 and 0.6, respectively. This understanding is crucial for data analysis and statistical modeling, particularly when dealing with normally distributed data sets.