Question

True/False: When plotted on the same graph, a distribution with a mean of 50 and a standard deviation of 10 will look more spread out than will a distribution with a mean of 60 and a standard deviation of \(5 .\)

Short answer

True, because the distribution with a standard deviation of 10 is more spread out.

Step-by-step solution

Understanding Distribution Spread

A distribution's spread is determined by its standard deviation. A larger standard deviation means the data is more spread out around the mean. Conversely, a smaller standard deviation indicates that data points are closer to the mean.

Analyzing Given Distributions

We have two distributions: Distribution A with a mean of 50 and standard deviation of 10, and Distribution B with a mean of 60 and standard deviation of 5. We will compare the standard deviations since the means do not influence the spread of data.

Comparing Standard Deviations

Distribution A has a standard deviation of 10, and Distribution B has a standard deviation of 5. Since 10 is greater than 5, Distribution A is more spread out than Distribution B.

Conclusion

Based on the comparison of standard deviations, Distribution A, which has a larger standard deviation, is more spread out on a graph compared to Distribution B. Therefore, the statement is True.

Key concepts

Distribution Spread

The concept of distribution spread is essential in understanding how data values are arranged around the mean in a dataset. This spread is quantitatively measured by the standard deviation. The larger the standard deviation, the more spread out the data points are. Conversely, if the standard deviation is small, most data points cluster closely around the mean.
For instance, imagine you toss two sets of balls towards a target. Set A lands closer together, and Set B scatters broadly. The spread of Set B is larger than Set A. Similarly, in statistical distributions, a larger spread indicates a greater diversity of values.
In the exercise example, Distribution A with a standard deviation of 10 has more spread compared to Distribution B with a standard deviation of 5. This is because 10 is greater than 5, indicating Distribution A is widely spread around its mean.

Mean and Standard Deviation Comparison

When analyzing distributions, it is crucial to understand the role of both the mean and the standard deviation. The mean provides a central value or an average of the dataset. However, while it tells us where most of the data points might be located on a graph, it doesn't tell us about the spread or variability of those data.

• Mean: Offers central tendency
• Standard Deviation: Measures spread or dispersion

In comparing the two distributions in the exercise, we focus on standard deviations because they tell us how spread out the data is, regardless of where the center (mean) is located. Distribution A (Std. Dev = 10) is spread wider than Distribution B (Std. Dev = 5), indicating that it covers more area around its mean of 50 than Distribution B does around its mean of 60.
Remember, while the mean shows position, the standard deviation indicates spread, and these combined give us a better understanding of a distribution's shape.

Distribution Graph Interpretation

Interpreting a distribution graph involves visualizing how data points fall in relation to the mean. When plotted, each distribution will have a curve, often a bell-shaped curve for normal distributions.
Distribution A and B, when plotted on the same graph, will showcase different spreads due to their differing standard deviations.

• Distribution A with a larger standard deviation will have a flatter and wider curve.
• Distribution B with a smaller standard deviation will appear narrow and steep.

By observing the graph, you can visually determine which distribution has more spread based on the curve's width and shape. A wider, more spread out curve represents a larger range of values, while a narrower curve indicates data points are clustered close to the mean.
Thus, visual interpretation reinforces the numerical understanding of standard deviation, confirming the exercise's conclusion: Distribution A is indeed more spread out than Distribution B.