Q1. Data: 8, 12, 15, 10, 5. Find the mean, variance, and standard deviation.

Background

Topic: Descriptive Statistics – Measures of Center and Spread

This question tests your ability to compute the sample mean, variance, and standard deviation for a small data set. These are foundational concepts in statistics for summarizing and understanding data.

Key Terms and Formulas

• Sample Mean:

• Sample Variance:

• Sample Standard Deviation:

Step-by-Step Guidance

1. Add all the data values together to find the sum .

2. Count the number of data points .

3. Calculate the mean using .

4. Subtract the mean from each data value, square the result, and sum these squared differences to get .

5. Divide this sum by to find the sample variance .

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

Try solving on your own before revealing the answer!

Q2. Given mean = 40, SD = 8, X = 56. Find the z-score.

Background

Topic: Standardization (Z-Score)

This question tests your ability to standardize a value using the z-score formula, which tells you how many standard deviations a value is from the mean.

Key Terms and Formula

• Z-score:

• = observed value, = mean, = standard deviation

Step-by-Step Guidance

1. Identify the observed value , mean , and standard deviation .

2. Subtract the mean from the observed value: .

3. Divide the result by the standard deviation: .

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Q3. Q1 = 22, Q3 = 50. Find the interquartile range (IQR).

Background

Topic: Measures of Spread – Interquartile Range

This question tests your understanding of quartiles and how to calculate the interquartile range, which measures the spread of the middle 50% of data.

Key Terms and Formula

• Interquartile Range (IQR):

• = first quartile (25th percentile), = third quartile (75th percentile)

Step-by-Step Guidance

1. Identify and from the problem: , .

2. Subtract from to find the IQR: .

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Q4. Mean = 100, SD = 20. Find the coefficient of variation (CV).

Background

Topic: Relative Measures of Spread

This question tests your ability to compute the coefficient of variation, which expresses the standard deviation as a percentage of the mean.

Key Terms and Formula

• Coefficient of Variation:

• Often expressed as a percentage:

Step-by-Step Guidance

1. Identify the standard deviation and mean .

2. Divide the standard deviation by the mean: .

3. If asked for a percentage, multiply the result by 100.

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Q5. r = -0.78. Interpret the strength and direction of the correlation.

Background

Topic: Correlation

This question tests your understanding of the correlation coefficient, which measures the strength and direction of a linear relationship between two variables.

Key Terms

• Correlation coefficient (): ranges from -1 to 1

• Strength: closer to -1 or 1 means stronger relationship

• Direction: positive () or negative ()

Step-by-Step Guidance

1. Look at the sign of . Negative means the relationship is negative (as one variable increases, the other decreases).

2. Assess the magnitude. is close to 1, indicating a strong relationship.

3. Combine both to interpret: strong negative linear relationship.

Try interpreting on your own before revealing the answer!