Question

After taking the state merit examinations for the positions of school counselor and social worker, you receive the following information on the tests and on your performance. On which of the tests did you do better? $$\begin{array}{rl}& \overline{\begin{array}{ll}\text{ School Counselor }& \text{ Social Worker }\\ \overline{X}=118& \overline{X}=27\\ s=17& s=3\\ \text{ Your score }=127& \text{ Your score }=29\end{array}}\\ & \begin{array}{rl}& \end{array}\end{array}$$

Short answer

You did better on the Social Worker test.

Step-by-step solution

- Understand the Problem

Determine on which test you performed better relative to others by comparing your scores to the averages and standard deviations of the tests.

- Standardize Scores

Convert your raw scores into z-scores to make a fair comparison between tests. Use the formula for the z-score: $$z=\frac{X-\overline{X}}{s}$$ where $X$ is your score, $\overline{X}$ is the average score, and $s$ is the standard deviation.

- Calculate Z-Score for School Counselor Test

Substitute the values into the z-score formula: $$X=127,\overline{X}=118,s=17$$ $$z_{\text{counselor}}=\frac{127-118}{17}=\frac{9}{17}\approx 0.53$$

- Calculate Z-Score for Social Worker Test

Substitute the values into the z-score formula: $$X=29,\overline{X}=27,s=3$$ $$z_{\text{worker}}=\frac{29-27}{3}=\frac{2}{3}\approx 0.67$$

- Compare Z-Scores

The z-score for the social worker test $0.67$ is higher than the z-score for the school counselor test $0.53$. This indicates that your performance on the social worker test is better relative to others than your performance on the school counselor test.

Key concepts

Standard Deviation

Understanding standard deviation is crucial for comparing test scores using z-scores. Standard deviation measures how spread out the scores are from the average (mean) score. It gives us an idea of the variability in the data.

A larger standard deviation means the scores vary widely from the mean. In contrast, a smaller standard deviation indicates that the scores are closer to the mean.

To compute the standard deviation, we use:

$$s=\sqrt{\frac{\mathrm{\Sigma }(X_i-\overline{X}{)}^2}{N-1}}$$

In this formula, $s$ denotes the standard deviation, $X_i$ represents each score, $\overline{X}$ is the mean, and $N$ is the number of scores. By understanding standard deviation, we grasp how your scores deviate from the average, which is essential for meaningful score comparisons.

Z-Score Calculation

Calculating z-scores is the next step in comparing your test performances. Z-scores standardize scores from different distributions, letting us compare them directly. We use the formula:

$$z=\frac{X-\overline{X}}{s}$$

Here, $X$ is your score, $\overline{X}$ is the mean score, and $s$ is the standard deviation. The z-score tells us how many standard deviations your score is from the mean:

• If the z-score is positive, your score is above the mean.
• If negative, your score is below the mean.
• A z-score of 0 means your score is exactly the average.

For example, to calculate the z-score for the school counselor test with $X=127$, $\overline{X}=118$, and $s=17$:

$$z_{\text{counselor}}=\frac{127-118}{17}=\frac{9}{17}\approx 0.53$$

Score Comparison

Once we have the z-scores, comparing test performances becomes straightforward. Z-scores allow us to see how well you did on each test relative to the average performance and the spread of scores.

In this exercise, we calculated two z-scores:

• School counselor test: z = 0.53
• Social worker test: z = 0.67

These z-scores show that you performed slightly better on the social worker test relative to others because the z-score of 0.67 is higher than 0.53.

A higher z-score indicates that your performance was further above the average, while a lower z-score would suggest you were closer to or below the mean. Understanding this helps you make informed decisions about your performances.

Statistical Analysis

Statistical analysis is the backbone of understanding and interpreting data effectively. By applying statistical methods like standard deviation and z-scores, we gain insights into how individual scores compare within a dataset.

For instance, in this exercise, we started by understanding the mean and standard deviation for each test. Then we calculated the z-scores, which transformed the raw scores into a standardized form for direct comparison.

Such analyses are vital in many real-world scenarios beyond test scores, including quality control in manufacturing, performance analysis in sports, and more. They allow us to make objective decisions based on numerical data and trends, rather than subjective judgments.

Thus, mastering basic statistical tools like mean, standard deviation, and z-scores empowers us to interpret data reliably and make data-driven decisions in various fields.