Algebra Test Scores

Getting Started

The student is unable to identify the correct distribution.

The student chooses the wrong distribution or leaves the paper blank.

What is different about the three distributions? Can you describe the shape of each distribution?

What kind of distribution will have 68% of the data within one standard deviation of the mean?

Review terminology to describe shapes of distributions to include symmetric, normal, skewed left, skewed right, and uniform. Be sure the student understands the 68-95-99.7 rule and that it only applies to normal distributions. Provide opportunities for the student to apply the 68-95-99.7 rule to determine the proportion of data within one, two, or three standard deviations of the mean of a normal distribution of data. Give the student a set of data whose distribution is normal along with its mean and standard deviation. Ask the student to calculate the values within one, two, or three standard deviations of the mean and then count the number of scores within these limits to confirm the 68-95-99.7 rule. Have the student engage in a similar exercise with a set of data whose distribution is not normal. Guide the student to compare the percentage of data within one, two, or three standard deviations of the mean to 68%, 95%, or 99.7% respectively, to observe the deviations from these percents. Frequently remind the student that the 68-95-99.7 rule only applies to normal distributions and other methods (such as the one described above) must be used to determine the proportion of data within a certain interval of other types of distributions.

Making Progress

The student identifies the correct distribution but cannot justify his or her answer or provides an incomplete justification.

The student chooses Distribution C but:

• Offers no explanation or justification.
• Uses incorrect terminology to describe Distribution C (e.g., as “regular” or “average”).
• Describes Distribution C as symmetric but does not use the term normal.

How would you describe the shape of the data for Distributions A and B? What terms are used to describe each of these distributions?

Can you justify why you chose Distribution C as appropriate? Why are Distributions A and B not appropriate?

Review terminology to describe shapes of distributions to include symmetric, normal, skewed left, skewed right, and uniform. Model explaining why it is only appropriate to conclude that 68% of the data is within ±1 standard deviation of the mean for Distribution C, emphasizing that Distribution C is approximately normal in shape. Also, explain why it is not appropriate to use this approach for Distributions A and B by referencing their shapes. Provide additional opportunities for the student to determine if the 68-95-99.7 rule can be used to determine the proportion of data within a certain interval of a distribution.

Got It

The student provides complete and correct responses to all components of the task.

The student chooses Distribution C and explains that this histogram approximates, in shape, a normal curve. Therefore, it is appropriate to apply the 68-95-99.7 rule.

What is the relationship between the mean and the median when you have a normal distribution? What happens to the mean and median when you have skewed data?

Provide a vertical scale for the graphs and ask the student to approximate the location of the middle 68% of the data in Distribution C (±1 standard deviation from the mean) by drawing vertical lines on the graph and estimating the lower and upper boundary of the test scores. Then ask the student to estimate the percentage of data between these boundaries for the other two distributions.