Q1. Interpreting Percentiles from an Ogive

Background

Topic: Percentiles and Ogives

This question tests your ability to read and interpret percentiles from an ogive (cumulative frequency graph). You are asked to determine what score corresponds to a specific percentile and to interpret what that percentile means in the context of the data.

Key Terms and Formulas:

• Percentile: The value below which a given percentage of observations in a group of observations falls.

• Ogive: A graph that represents the cumulative frequencies for the classes in a frequency distribution.

Step-by-Step Guidance

1. Identify the percentile you are interested in (for example, the 60th percentile).

2. On the ogive, locate the percentile value on the vertical axis (y-axis).

3. Draw a horizontal line from this percentile value until it meets the ogive curve.

4. From the point of intersection, draw a vertical line down to the horizontal axis (x-axis) to find the corresponding number of goals.

5. Interpret the meaning: What does it mean for a team to be at or below this percentile in terms of goals scored?

Try solving on your own before revealing the answer!

Q2. Effects of Outliers on Mean and Median

Background

Topic: Measures of Center (Mean and Median) and Outliers

This question asks you to calculate the mean and median of a data set, both with and without an outlier, and to interpret how the outlier affects these measures.

Key Terms and Formulas:

• Mean:

• Median: The middle value when the data are arranged in order.

• Outlier: A value much higher or lower than most of the other values in a data set.

Step-by-Step Guidance

1. List the data in order: 3, 12, 18, 22, 28, 30, 34, 40, 48, 52, 170.

2. Calculate the mean and median for the full data set (including the outlier 170).

3. Remove the outlier (170) and recalculate the mean and median for the remaining 10 values.

4. Compare the results: How did the mean and median change after removing the outlier?

5. Interpret which measure (mean or median) is more resistant to outliers and why.

Try solving on your own before revealing the answer!

Q3. Calculating the Geometric Mean of Growth Rates

Background

Topic: Geometric Mean

This question asks you to find the single percentage growth rate that is equivalent to several consecutive growth rates by calculating the geometric mean.

Key Terms and Formulas:

• Geometric Mean:

• Convert percentages to decimals before multiplying (e.g., 1.2% becomes 1.012).

Step-by-Step Guidance

1. Convert each percentage growth rate to its decimal multiplier (e.g., 1.2% becomes 1.012).

2. Multiply all the decimal multipliers together.

3. Take the nth root of the product, where n is the number of years (in this case, 5).

4. Convert the result back to a percentage by subtracting 1 and multiplying by 100.

Try solving on your own before revealing the answer!

Q4. Range, Variance, and Standard Deviation of Workout Durations

Background

Topic: Measures of Variability (Range, Variance, Standard Deviation)

This question asks you to calculate the range, variance, and standard deviation for a sample data set and interpret the results.

Key Terms and Formulas:

• Range:

• Sample Variance:

• Sample Standard Deviation:

Step-by-Step Guidance

1. List the data: 30, 45, 60, 50, 40, 55, 70, 35, 65, 50, 42.

2. Find the maximum and minimum values to calculate the range.

3. Calculate the mean () of the data set.

4. Compute the squared differences from the mean for each value, sum them, and divide by (n-1) to get the variance.

5. Take the square root of the variance to find the standard deviation.

6. Interpret what the values of range, variance, and standard deviation tell you about the spread of the data.

Try solving on your own before revealing the answer!

Q5. Drawing and Interpreting a Box-and-Whisker Plot

Background

Topic: Boxplots (Box-and-Whisker Plots)

This question asks you to select the correct boxplot for a given data set and understand how to interpret the five-number summary (minimum, Q1, median, Q3, maximum).

Key Terms and Formulas:

• Boxplot: A graphical representation of the five-number summary of a data set.

• Five-number summary: Minimum, Q1 (first quartile), Median (Q2), Q3 (third quartile), Maximum.

Step-by-Step Guidance

1. Order the data set from smallest to largest.

2. Find the minimum, Q1, median, Q3, and maximum values.

3. Match these values to the features of each boxplot option.

4. Check which boxplot correctly represents the spread and center of the data.

Try solving on your own before revealing the answer!

Q6. Measures of Central Tendency and the Effect of Trimming

Background

Topic: Measures of Central Tendency (Mean, Median, Mode, Midrange) and Trimmed Data

This question asks you to calculate and compare the mean, median, mode, and midrange for both the original and a trimmed data set.

Key Terms and Formulas:

• Mean:

• Median: The middle value when the data are ordered.

• Mode: The value(s) that appear most frequently.

• Midrange:

• Trimmed Mean: The mean after removing a certain percentage of the lowest and highest values.

Step-by-Step Guidance

1. Order the data: 7, 9, 11, 12, 14, 15, 17, 18, 20, 21.

2. Calculate the mean, median, mode, and midrange for the original data set.

3. Trim 10% from each end (remove the lowest and highest value), then recalculate the measures for the trimmed data set.

4. Compare the results and interpret how trimming affects each measure.

Try solving on your own before revealing the answer!

Q7. Interpreting Standard Deviation in Bonus Distributions

Background

Topic: Standard Deviation and Probability

This question asks you to use the mean and standard deviation to determine which company is more likely to give a bonus of $6,000 or higher, given the average and standard deviation of bonuses at two companies.

Key Terms and Formulas:

• Mean: The average value.

• Standard Deviation: A measure of how spread out the values are around the mean.

• Empirical Rule (68-95-99.7 Rule): For a normal distribution, about 68% of values fall within one standard deviation of the mean.

Step-by-Step Guidance

1. Note the mean ($5,000) and standard deviations ($800 for X, $500 for Y).

2. Calculate how many standard deviations above the mean $6,000 is for each company.

3. Recall that a lower standard deviation means values are more tightly clustered around the mean.

4. Compare the likelihood of receiving a bonus of $6,000 or higher at each company based on the standard deviation.

Try solving on your own before revealing the answer!

Q8. Finding Quartiles from a Data Set

Background

Topic: Quartiles (Q1, Q2, Q3)

This question asks you to find the first quartile (Q1), median (Q2), and third quartile (Q3) for a given data set.

Key Terms and Formulas:

• Quartiles: Values that divide a data set into four equal parts.

• Q1: The median of the lower half of the data.

• Q2: The median of the data set.

• Q3: The median of the upper half of the data.

Step-by-Step Guidance

1. Order the data: 7, 9, 9, 9, 10, 11, 12, 12, 12, 14, 14, 15, 17, 20.

2. Find the median (Q2) of the data set.

3. Find Q1 by taking the median of the lower half (not including Q2 if the number of data points is odd).

4. Find Q3 by taking the median of the upper half.

Try solving on your own before revealing the answer!