Step-by-Step Guidance for Descriptive Statistics: Range, Median, Mode, Mean, Variance, and Standard Deviation

Q1. For the following data set, find the: i) Range ii) Median iii) Mode iv) Mean v) Sample Variance vi) Sample Standard Deviation vii) Population Variance viii) Population Standard Deviation Data set: 1, 3, 5, 7, 3, 3, 2, 7, 8, 2, 4, 9, 10

Background

Topic: Descriptive Statistics (Measures of Central Tendency and Dispersion)

This question tests your ability to compute and interpret basic descriptive statistics for a data set, including measures of center (mean, median, mode) and measures of spread (range, variance, standard deviation) for both samples and populations.

Key Terms and Formulas

• Range: The difference between the largest and smallest values in the data set. $\text{Range} = \text{Maximum} - \text{Minimum}$

• Median: The middle value when the data are ordered from least to greatest.

• Mode: The value(s) that appear most frequently in the data set.

• Mean (Sample): The average of the data values. $\bar{x} = \frac{\sum x_i}{n}$

• Sample Variance: Measures the average squared deviation from the mean (for a sample). $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$

• Sample Standard Deviation: The square root of the sample variance. $s = \sqrt{s^2}$

• Population Variance: Measures the average squared deviation from the mean (for a population). $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

• Population Standard Deviation: The square root of the population variance. $\sigma = \sqrt{\sigma^2}$

Step-by-Step Guidance

1. Order the data from smallest to largest. This will help you find the minimum, maximum, median, and mode more easily.

2. Find the range: Identify the smallest and largest values in the ordered data set, then subtract the minimum from the maximum using the range formula above.

3. Find the median: Count the number of data points ($n$). If $n$ is odd, the median is the middle value. If $n$ is even, the median is the average of the two middle values.

4. Find the mode: Determine which value(s) appear most frequently in the data set.

5. Calculate the mean: Add up all the data values and divide by the number of data points ($n$).

6. Set up the computation table for variance and standard deviation: For each data value, subtract the mean and square the result. Sum these squared differences. For sample variance, divide by $n-1$; for population variance, divide by $n$.

Try solving on your own before revealing the answer!

Final Answers:

• Range: 10 - 1 = 9

• Median: 4

• Mode: 3

• Mean: 5

• Sample Variance: 8.08

• Sample Standard Deviation: 2.84

• Population Variance: 7.48

• Population Standard Deviation: 2.73

Each value was calculated using the appropriate formula and the ordered data set. Remember to use $n-1$ for sample variance and $n$ for population variance.

Q2. For the following data set, find the: i) Range ii) Median iii) Mode iv) Mean v) Sample Variance vi) Sample Standard Deviation vii) Population Variance viii) Population Standard Deviation Data set: 0, 6, -3, 5, -12, 4, 17, 3, -9, 10, -3, 2, 0

Background

Topic: Descriptive Statistics (Measures of Central Tendency and Dispersion)

This question is similar to Q1, but with a data set that includes negative values. The process for finding each statistic remains the same.

Key Terms and Formulas

• See the formulas and definitions listed in Q1 above.

Step-by-Step Guidance

1. Order the data from smallest to largest to identify the minimum, maximum, and to help with median and mode.

2. Find the range by subtracting the minimum value from the maximum value.

3. Find the median by determining the middle value (since $n$ is odd, it's the value at position $(n+1)/2$ in the ordered list).

4. Find the mode by identifying the value(s) that appear most frequently.

5. Calculate the mean by summing all values and dividing by $n$.

6. Set up the computation table for variance and standard deviation as in Q1.

Try solving on your own before revealing the answer!

Final Answers:

• Range: 17 - (-12) = 29

• Median: 2

• Mode: -3 and 0 (bimodal)

• Mean: 1.23

• Sample Variance: 70.98

• Sample Standard Deviation: 8.43

• Population Variance: 65.38

• Population Standard Deviation: 8.09

Negative values are handled the same way as positive values in all calculations.

Q3. For the following data set, find the: i) Range ii) Median iii) Mode iv) Mean v) Sample Variance vi) Sample Standard Deviation vii) Population Variance viii) Population Standard Deviation Data set: 1.1, 1.2, 1.2, 1.3, 1.2, 1.1, 2.2, 1.5, 1.6, 1.4, 1.7, 1.4, 1.9

Background

Topic: Descriptive Statistics (Measures of Central Tendency and Dispersion)

This question uses a data set with decimal values. The process for finding each statistic is the same as in previous questions.

Key Terms and Formulas

• See the formulas and definitions listed in Q1 above.

Step-by-Step Guidance

1. Order the data from smallest to largest to help with all calculations.

2. Find the range by subtracting the smallest value from the largest value.

3. Find the median by locating the middle value in the ordered list.

4. Find the mode by identifying the value(s) that appear most frequently.

5. Calculate the mean by summing all values and dividing by $n$.

6. Set up the computation table for variance and standard deviation as in previous questions.

Try solving on your own before revealing the answer!

Final Answers:

• Range: 2.2 - 1.1 = 1.1

• Median: 1.4

• Mode: 1.2

• Mean: 1.49

• Sample Variance: 0.13

• Sample Standard Deviation: 0.36

• Population Variance: 0.12

• Population Standard Deviation: 0.35

Be careful with decimal arithmetic and double-check your calculations for accuracy.

Q4. For the following data set, find the: i) Range ii) Median iii) Mode iv) Mean v) Sample Variance vi) Sample Standard Deviation vii) Population Variance viii) Population Standard Deviation Data set: 25, 25, 24, 24, 16, 16, 16, 23, 24, 25, 27, 28, 29, 31, 32, 34, 34, 12, 15, 17, 15, 15

Background

Topic: Descriptive Statistics (Measures of Central Tendency and Dispersion)

This question uses a larger data set with repeated values. The process for finding each statistic is the same as in previous questions.

Key Terms and Formulas

• See the formulas and definitions listed in Q1 above.

Step-by-Step Guidance

1. Order the data from smallest to largest to help with all calculations.

2. Find the range by subtracting the smallest value from the largest value.

3. Find the median by locating the middle value(s) in the ordered list (since $n$ is even, average the two middle values).

4. Find the mode by identifying the value(s) that appear most frequently.

5. Calculate the mean by summing all values and dividing by $n$.

6. Set up the computation table for variance and standard deviation as in previous questions.

Try solving on your own before revealing the answer!

Final Answers:

• Range: 34 - 12 = 22

• Median: 24.5

• Mode: 15

• Mean: 23.09

• Sample Variance: 44.18

• Sample Standard Deviation: 6.65

• Population Variance: 42.18

• Population Standard Deviation: 6.50

With a larger data set, it's especially important to organize your work and double-check each step.