Problem 78 - 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. \(-1\) standard deviation

Step-by-Step Solution

Answer

The data value which is \(-1\) standard deviation away from the mean is 32.8.

Step 1: Understand the problem

We are given a mean of 39 and a standard deviation of 6.2 and asked to find the value that is \(-1\) standard deviation away from the mean. Remember: \(1\) standard deviation below the mean can be calculated by subtracting \(1\) times the standard deviation from the mean.

Step 2: Calculate the value

So, to find the value, we can use the formula \(mean - (number \, of \, standard \, deviations \times standard \, deviation)\). Plugging in our values, we get \(39 - (1 \times 6.2) = 32.8\).

Key Concepts

Mean and Standard Deviation

The **mean** is a measure of central tendency, which tells us the average value of a data set. Think of the mean as the "level ground" from which we can measure other points. In our example, the mean is 39. This means if you add up all the data points in our set and divide by the number of points, each would average out to 39.

On the other hand, the **standard deviation** speaks to the spread or variation of data points around this mean. A smaller standard deviation implies that the data points are closely packed around the mean, while a larger one suggests a wider spread. Here, the standard deviation is 6.2, indicating that most data points will cluster within 6.2 units from the mean, whether above or below. This allows us to understand not just the center of the data set but also how "spread out" or variable the data is.

Subtracting Standard Deviations

When we're asked to find a value that is a certain number of standard deviations away from the mean, it often involves simple arithmetic. If we need to find a value that is -1 standard deviation from the mean, we subtract one standard deviation from the mean value. Using our exercise example: \( mean - (standard\, deviation \times number\, of\, deviations) \)

Place the formula parameters:

• **Mean** = 39
• **Standard deviation** = 6.2
• **Number of standard deviations** = -1 (because we are moving down)

Solution: \( 39 - (1 \times 6.2) = 32.8 \)

Thus, 32.8 is exactly one standard deviation below the mean, showcasing how subtraction is used to navigate the distance on a normal distribution curve.

Data Set Analysis

Analyzing a data set involves understanding how its elements relate to each other through measures like the mean and standard deviation. A key part of this is interpreting how different data points fall relative to these measures.

In a **normally distributed** set, most data points will cluster around the mean. To visualize, imagine the familiar bell curve representing a normal distribution curve:

• **Mean**: The peak or the center of the curve.
• **Standard deviation**: Dictates the steepness or flatness of the bell. The lower the standard deviation, the steeper the bell.
• Data points: Most (about 68%) will be within one standard deviation away, either above or below from the mean.

Understanding this helps predict where most of the data will lie, enabling predictions about typical values and the likelihood of extreme cases appearing in the data set.