Problem 77 - Algebra 2

Question

A set of data with a mean of 39 and a standard deviation of 6.2 is normally distributed. Find each value, given its distance from the mean. \(+3\) standard deviations

Step-by-Step Solution

Answer

The value that is +3 standard deviations from the mean is 57.6.

Step 1: Understanding the problem

In this data set, the mean is 39, and the standard deviation is 6.2. We are asked to find the value that is +3 standard deviations away from the mean. The '+' sign indicates that the value is above the mean. This means, for each standard deviation we add to the mean, we are moving to the right on the number line. That is, we are looking for a value higher than the mean.

Step 2: Calculating the Value

To calculate the value, we simply add three times the standard deviation to the mean. So, the calculation would be Mean + 3*(Standard Deviation), which is \(39 + 3*6.2\). Doing the multiplication first (because of the rule of BIDMAS/BODMAS), we get \(39 + 18.6\).

Step 3: Final Calculation

Adding these together, we get the final result of \(57.6\). Therefore, the value that is three standard deviations away from the mean in the positive direction is 57.6.

Key Concepts

Mean

The mean, also known as the average, is a measure of central tendency. It's a way to find the "center" or "middle" value of a data set. To calculate the mean, simply add up all the numbers in the set and then divide by the number of values.
For example, if you have a data set with values like 2, 3, 5, 7, and 11, you first add them to get 28. Then, you divide by the number of values, which is 5.

• Mean = (2 + 3 + 5 + 7 + 11) / 5 = 28 / 5 = 5.6

The mean provides a quick summary of the data set. However, it doesn't show how spread out the numbers are. That's where standard deviation comes into play.

Standard Deviation

Standard deviation is a statistic that represents the degree to which individual data points in a set deviate from the mean. A higher standard deviation indicates that values are spread out over a wider range, while a lower standard deviation shows that the values are closer to the mean.
To calculate standard deviation manually, follow these steps:

• Find the mean of the data set.
• Subtract the mean from each data point and square the result.
• Find the average of these squared differences.
• Take the square root of this average.

Calculating standard deviation helps in understanding how variable the dataset is in comparison to the mean.

Z-score

A z-score, or standard score, measures the number of standard deviations a data point is from the mean. It tells you where the data point is in relation to the average, helping you determine if it is typical or unusual.
The formula for calculating a z-score is:

• \[ z = \frac{x - \text{mean}}{\text{standard deviation}} \]

Where:

• \(x\) is the data point.
• 'mean' is the average of the dataset.
• 'standard deviation' is how much variation exists from the average.

Understanding z-scores is crucial in determining how likely a data point is, statistically, in a distribution.

BIDMAS/BODMAS

BIDMAS/BODMAS are acronyms that represent the order of operations in mathematics. They help solve equations correctly by following a specific sequence:

• B for Brackets
• I/D stands for Indices or Orders (exponents, powers, roots)
• D for Division and M for Multiplication (from left to right)
• A for Addition and S for Subtraction (from left to right)

For example, in solving the equation \(2 + 3 \times 4\), you first do the multiplication: \(3 \times 4 = 12\), then add 2 to get 14. Remembering BIDMAS/BODMAS ensures that calculations are structured and accurate.