91Ó°ÊÓ

Standard deviation is a critical concept in statistics, particularly when dealing with a normal distribution. It is a measure that quantifies the amount of variation or dispersion of a set of values. In simpler terms, it indicates how spread out the numbers in a dataset are around the mean (average).
In a normal distribution, most data points fall close to the mean, and the standard deviation helps to determine how spread out these points are. A smaller standard deviation means that the values tend to be close to the mean, while a larger standard deviation indicates that the numbers are more spread out.
For example, if we have a mean spending of \(6312 as in our problem and we find a standard deviation of around \)3221.64, it tells us that the spending amounts typically vary around this mean by that amount. To calculate the standard deviation when you know the z-score and a data point as in the exercise:

• Identify your z-score, which indicates the number of standard deviations a data point is from the mean.
• Use the formula \( z = \frac{x - \mu}{\sigma} \) where \( x \) is your data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• Rearrange to solve for \( \sigma \).

This calculation allows us to understand how variable the transportation spending data is.

A z-score, also known as a standard score, is essentially a numerical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean.
Calculating a z-score is an integral part of the normal distribution, as it helps standardize different datasets making analysis comparative and comprehensible. It can tell you exactly how many standard deviations a value is from the mean. In our transportation spending example, the z-score helps determine how unusual a specific spending amount is compared to the average.Here's how to calculate a z-score:

• Subtract the mean from your data point.
• Divide the result by the standard deviation: \( z = \frac{x - \mu}{\sigma} \).

The resulting z-score indicates its positioning on the normal distribution curve. For example, a z-score of 0 indicates the data point is exactly at the mean, while a positive or negative z-score shows it is above or below the mean, respectively.When it comes to probabilities, the z-score can also be used to find the percentage of values below, above, or between data points in a normal distribution by referring to standard normal distribution tables or using statistical software.

Percentiles are a statistical concept used to understand and interpret data distributions. They rank data points to indicate the percentage of values within a dataset that lie below a specific point. Understanding percentiles is crucial for grasping how a particular data point compares within the entire dataset.
In simple terms, the nth percentile is the value below which n% of the data falls. For instance, being in the 97th percentile means a data point is greater than or equal to the value of 97% of the remaining data. To find the percentile in a normal distribution:

• Identify the percentile position you are interested in (e.g., the top 3%, which corresponds to the 97th percentile).
• Use a standard normal distribution table or calculator to find the corresponding z-score for the desired percentile.
• Using the z-score, calculate the value (data point) using \( x = \mu + z \sigma \).

For instance, in our problem, the 97th percentile calculation for spending helped us find that the top 3% of households spend more than $12374.1. Understanding percentiles can be vital in making informed decisions based on how extreme or typical a data point is within its distribution.