Problem 10 - Algebra 2

Question

A set of data with a mean of 62 and a standard deviation of 5.7 is normally distributed. Find each value, given its distance from the mean. \(-1\) standard deviation

Step-by-Step Solution

Answer

The value which is at one standard deviation below the mean of the given normal distribution is 56.3

Step 1: Understanding normal distribution

Normal distribution is a type of continuous probability distribution for a real-valued random variable. The data in such distribution is centered around the mean. The 'standard deviation' determine the spread of the data. A point located at 1 standard deviation away from the mean on the distribution curve covers approximately 68% of the data, 2 standard deviations cover approximately 95% and 3 standard deviations cover about 99.7%.

Step 2: Calculating value for -1 Standard deviation

In a normal distribution, when a value is -1 standard deviation away from the mean, it means it is 1 standard deviation below the mean. We can find this value by subtracting 1 standard deviation from the mean. The given mean, \(\mu\), is 62 and the standard deviation, \(\sigma\), is 5.7. So, to calculate the value for -1 standard deviation away from the mean, we use the formula: Value = \(\mu - \sigma\)

Step 3: Substitute the given values

Now substituting the given values in our formula, we would have: Value = 62 - 5.7.

Step 4: Calculating the Result

By subtracting 5.7 from 62, the obtained value is 56.3

Key Concepts

Understanding Standard Deviation

Standard deviation is a key concept in statistics that measures how spread out the numbers are in a set of data. It is a critical indicator of the variability or dispersion from the mean. Imagine it as a way to see how much the data "hug" or "stray" from the average value.

### Calculating Standard Deviation There are three steps involved when calculating standard deviation:

• Determine the mean, or average, of the data set.
• Subtract the mean from each data point and square the result.
• Find the average of these squared differences, and then take the square root of that average.

This calculation gives a numeric value that helps quantify how varied the individual numbers of a data set are from the mean. A larger standard deviation signals more spread out data, while a smaller one indicates data points that are closer to the mean.

Grasping the Concept of Mean

The mean, often known as the average, is the sum of all numbers in a data set divided by the total number of values in that set. It's one of the simplest measures of central tendency, providing a single number that summarizes an entire data set.

To calculate the mean:

• Add up all the numbers in your data set.
• Count how many numbers are in the set.
• Divide the total sum by the number of data points.

For example, if your data set is \(4, 8, 6, 5\), the mean would be calculated as \((4 + 8 + 6 + 5)/4 = 5.75\). The mean provides a simple snapshot of the overall data, which is why it's so widely used.

Navigating Probability Distribution

Probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. There are various types of distributions, with the normal distribution being one of the most significant in statistics.

In a normal distribution, data is symmetrically distributed with most of the observations clustering around the central peak, known as the mean. The tails on either end of the distribution curve represent extreme values away from the mean. This bell-shaped curve is characterized by two parameters: the mean and the standard deviation.

### Features of Normal Distribution Normal distribution has several notable properties:

• The area under the curve totals 1, representing the entire probability of all outcomes.
• The mean, median, and mode of a normal distribution are all the same.
• It's defined by its mean and standard deviation, which dictates the width and height of the curve.

In practical applications, normal distribution is pivotal because it fits numerous natural phenomena. It is instrumental in inferential statistics and hypothesis testing, allowing predictions about a population based on sample data.