Question

Compute the range and standard deviation of the following 10 test scores. $$ 66,75,69,72,84,90,96,70,55,45 $$

Short answer

Range = 51, Standard Deviation ≈ 14.56

Step-by-step solution

- Arrange the Data

First, arrange the test scores in ascending order: 45, 55, 66, 69, 70, 72, 75, 84, 90, 96

- Calculate the Range

The range is the difference between the highest and lowest scores. Subtract the smallest value (45) from the largest value (96): Range = 96 - 45 = 51

- Find the Mean

Calculate the mean (average) of the test scores by summing all the values and dividing by the number of scores. Sum = 45 + 55 + 66 + 69 + 70 + 72 + 75 + 84 + 90 + 96 = 722 Mean = \( \frac{722}{10} = 72.2 \)

- Calculate Each Score’s Deviation from the Mean

Subtract the mean from each score to find the deviation: 45 - 72.2 = -27.2 55 - 72.2 = -17.2 66 - 72.2 = -6.2 69 - 72.2 = -3.2 70 - 72.2 = -2.2 72 - 72.2 = -0.2 75 - 72.2 = 2.8 84 - 72.2 = 11.8 90 - 72.2 = 17.8 96 - 72.2 = 23.8

- Square Each Deviation

Square each of the deviations to make them positive: (−27.2)^2 = 739.84 (−17.2)^2 = 295.84 (−6.2)^2 = 38.44 (−3.2)^2 = 10.24 (−2.2)^2 = 4.84 (−0.2)^2 = 0.04 2.8^2 = 7.84 11.8^2 = 139.24 17.8^2 = 316.84 23.8^2 = 566.44

- Find the Mean of These Squared Deviations

Sum all the squared deviations and divide by the number of scores: Sum of Squared Deviations = 739.84 + 295.84 + 38.44 + 10.24 + 4.84 + 0.04 + 7.84 + 139.24 + 316.84 + 566.44 = 2119.6 Variance (σ^2) = \( \frac{2119.6}{10} = 211.96 \)

- Calculate the Standard Deviation

Take the square root of the variance to find the standard deviation: Standard Deviation = \( \frac{211.96}{10} = 14.56 \)

Key concepts

Understanding Test Scores

Test scores are numerical values that represent a student's performance on an exam or assessment. These scores provide a quantitative measure of how well a student has understood the material.
However, looking at individual test scores can be misleading. To get a better sense of overall performance and comparison between students, we use statistical tools like range and standard deviation.
This helps in understanding if the scores are closely grouped or widely dispersed.

Introduction to Statistical Analysis

Statistical analysis is a branch of mathematics dealing with data collection, interpretation, and presentation. It allows us to extract meaningful insights and make decisions based on data. Statistical analysis can show patterns, trends, and relationships.
In our example, we'll use statistical measures to analyze test scores. First, we organize the data by arranging the scores in ascending order. This step is crucial for calculating other measures like the range.
Statistical analysis often involves measures of central tendency, like the mean, and measures of variability or dispersion, like the range and standard deviation.
These tools help us understand not just the average performance but also the spread of the scores.

Variance and Deviation Explained

Variance and deviation are two key concepts in statistics used to describe the spread of a set of data points. Variance measures how far each data point is from the mean. Specifically, it's the average of the squared differences from the mean.
In our exercise, we calculated the variance by first finding each score's deviation from the mean and then squaring each of these deviations. The squared deviations are then summed up and divided by the number of data points to find the variance.
Standard deviation, on the other hand, is the square root of the variance, and it provides a measure of dispersion in the same unit as the data. It's a more intuitive measure compared to variance because it tells us, on average, how much each score differs from the mean.
A low standard deviation indicates that the data points are close to the mean, whereas a high standard deviation signifies data points that are spread out over a larger range of values.
By understanding and calculating these measures, we get a clearer picture of the variability in test scores, allowing educators to identify patterns and potentially address areas needing improvement.