Question

The mean score on the Stats exam was 75 points with a standard deviation of 5 points, and Gregor's z-score was \(-2\). How many points did he score?

Short answer

Gregor scored 65 points.

Step-by-step solution

Understanding the Formula for Z-score

The z-score is calculated using the formula \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the individual score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Our goal is to find \( X \).

Substitute the Known Values into the Formula

We have a z-score \( z = -2 \), mean \( \mu = 75 \), and standard deviation \( \sigma = 5 \). Substitute these values into the z-score formula: \( -2 = \frac{X - 75}{5} \).

Solve for the Score X

Rearrange the equation to solve for \( X \). First, multiply both sides by 5: \( -2 \times 5 = X - 75 \). This results in \( -10 = X - 75 \).

Isolate X

Add 75 to both sides of the equation to solve for \( X \): \( X = -10 + 75 \). Simplifying gives \( X = 65 \).

Key concepts

Standard Deviation

Standard deviation is a crucial concept in statistics that helps us understand data dispersion. It's a measure that indicates how much individual data points in a dataset differ from the mean, or average. In simpler terms, it tells us how spread out the numbers are. A low standard deviation means that most numbers are close to the mean. Conversely, a high standard deviation indicates numbers are spread out over a wide range.

In our scenario, the standard deviation for the exam scores is 5 points. This means that most of the scores were within 5 points from the average score of 75. By understanding standard deviation, you can better grasp how individual scores compare to the average of a dataset. It's like getting a sense of the 'typical' variance from the mean! This concept is vital for calculating z-scores, as it reflects the variability of the entire dataset.

Mean Score

The mean score is simply the average score obtained in a data set, and it serves as a central point for understanding the data distribution. To calculate the mean, you add up all the individual scores and then divide by the number of scores. In statistical terms, it's denoted by the symbol \( \mu \).

For the Stats exam, the mean score is given as 75. This tells us that if we add all the scores and divide by the total number of students, the average per student would be 75 points. The mean provides a benchmark for comparing each individual score. It gives a sense of the overall performance of students on the exam. Knowing the mean is imperative when calculating a z-score, as it is part of the formula to evaluate how far or close a score is relative to the mean.

Individual Score

An individual score refers to a specific data point within a dataset. In our context, it's the score that a particular student, Gregor, received on the Stats exam. Calculating the individual score involves using elements like the z-score, mean, and standard deviation, to determine where it stands in relation to the whole dataset.

In the problem, to find Gregor's score, we rearrange the z-score formula to solve for his individual score, denoted by \( X \). We use the given z-score, mean, and standard deviation. After doing the calculations, it turns out Gregor scored 65 points on his exam. Understanding how to find an individual score is quite helpful, particularly when you have a z-score. It allows you to pinpoint exactly where a single data point (or score) lies in the whole data set, whether it's below or above the average.

Standardized Score

A standardized score, or z-score, is a statistical measurement that describes a score's relationship to the mean of a group of scores. It's expressed in terms of standard deviations from the mean. The z-score indicates how many standard deviations an element is from the mean.

For Gregor, the z-score is \(-2\). This tells us that his score is 2 standard deviations below the mean score of 75 points. To figure out Gregor's actual score, you can manipulate the z-score formula, \( z = \frac{X - \mu}{\sigma} \), where \( X \) is the score we're solving for. This use of standardized scores helps us easily compare data points across different distributions, making it an invaluable tool in statistics. It's like having a universal language for scores, where you can understand the context of any score regardless of its scale.