Problem 38 - Thinking Mathematically

Question

Explain how to use a table showing \(z\)-scores and percentiles to determine the percentage of data items between two \(z\)-scores.

Step-by-Step Solution

Answer

The percentage of data items between two z-scores is calculated by determining the corresponding percentiles for both z-scores from the z-score table and subtracting the percentile of the lower z-score from the percentile of the higher z-score. The result is then multiplied by 100 to give a percentage.

Step 1: Understand z-Scores

A z-score is a statistical measurement that describes a value's relationship to the mean (average) of a group of values. If a z-score is 0, it indicates that the data point's score is identical to the mean score. A z-score of 1.0 indicates a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

Step 2: Using the Z-Score Table

Locate the z-scores on the z-score table. The table will provide the area to the right under the standard normal curve. Keep in mind that the table will usually contain only positive z-scores.

Step 3: Calculate the Difference

To find the percentage of data items between the two z-scores, locate the corresponding percentiles in the z-score table for both z-scores. Subtract the percentile of the lower z-score from the percentile of the higher z-score.

Step 4: Convert to Percentage

The resultant value is the proportion of data items between the two z-scores. To convert this value to a percentage, multiply the value by 100.

Key Concepts

Percentiles

Percentiles are essential in statistics for understanding the distribution of data. A percentile indicates the relative standing of a value within a dataset. For instance, the 90th percentile means that 90% of the data points are below this value. Understanding percentiles is crucial when working with z-scores. In a normal distribution, percentiles help us see where a particular score stands compared to the entire dataset. When you look at a z-score table, you can find the area under the curve that corresponds to each z-score, which tells you what percentile a certain score is at. Percentiles give a clear idea of how a particular score compares to the rest of the dataset, which makes them a handy tool for deciding on cut-off points in standardized testing or determining eligibility criteria in various applications.

• Percentile rank shows the percentage of scores that fall below a particular value.
• Useful for comparing scores from different datasets or tests.
• Helps to understand and interpret data distributions effectively.

Standard Deviation

Standard deviation is a measure that indicates the spread or variability of a dataset. It tells you how much individual data points differ from the mean of the dataset. A low standard deviation suggests that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values. In the context of z-scores, standard deviation is used to express how far a particular score is from the mean. A z-score of 1 means the score is one standard deviation above the mean, and a z-score of -1 means it is one standard deviation below the mean. Standard deviation provides insights into the consistency of a dataset, helping us understand the typical deviation of data points and thus the reliability of the mean as a measure of central tendency.

• Indicates data variability relative to the mean.
• Helps assess the consistency or spread in a dataset.
• Useful in determining how unusual or typical a particular data point is compared to others.

Normal Distribution

Normal distribution, often known as the bell curve, is a common continuous probability distribution that is symmetrical around its mean. Most of the data points lie close to the mean making the curve bell-shaped. A key characteristic of normal distribution is that it is fully defined by its mean and standard deviation. About 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. Z-scores are intimately related to normal distribution because they help standardize scores on a normal curve. This means that any dataset that approximates a normal distribution can be analyzed using z-scores to find relative positions within the dataset. Using a z-score table with a normal distribution assists in determining the percentage of the data between different z-scores. This helps in visualizing the proportion of scores that fall within specified ranges, aiding in various statistical analyses and decision-making processes.

• Symmetrical around the mean, resembling a bell shape.
• Most data falls close to the mean, illustrating the data's central tendency.
• Defined by mean and standard deviation, crucial for statistical interpretation.