Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 25m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

3. Describing Data Numerically

Median

Statistics

3. Describing Data Numerically

Median

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2:55 minutes

Problem 2.T.1a

Textbook Question

The overall averages of 12 students in a statistics class prior to taking the final exam are listed.
67 72 88 73 99 85 81 87 63 94 68 87

a. Find the mean, median, and mode of the data set. Which best represents the center of the data?

Verified step by step guidance

Step 1: To find the mean, calculate the sum of all the data points and divide by the total number of data points. Use the formula:

Mean = \sum x

_in, where

\sum x

_i is the sum of all values and

n

is the number of values.

Step 2: To find the median, first arrange the data in ascending order: 63, 67, 68, 72, 73, 81, 85, 87, 87, 88, 94, 99. Since there are 12 data points (an even number), the median is the average of the 6th and 7th values in the ordered list. Use the formula:

Median = x

₆+x₇2.

Step 3: To find the mode, identify the value(s) that appear most frequently in the data set. In this case, check the frequency of each value in the list: 63, 67, 68, 72, 73, 81, 85, 87, 87, 88, 94, 99.

Step 4: Compare the mean, median, and mode to determine which best represents the center of the data. Consider the distribution of the data (e.g., whether it is symmetric, skewed, or has outliers) to decide which measure of central tendency is most appropriate.

Step 5: Conclude which measure (mean, median, or mode) is the best representation of the center of the data based on the context and the characteristics of the data set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean

The mean is the average of a data set, calculated by summing all the values and dividing by the number of values. It provides a central value that represents the overall data but can be influenced by extreme values (outliers). In the context of the given student scores, calculating the mean will give a quick overview of their performance.

Median

The median is the middle value of a data set when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers. It is a robust measure of central tendency that is less affected by outliers, making it useful for understanding the typical performance of the students.

Mode

The mode is the value that appears most frequently in a data set. It can be particularly useful in identifying the most common score among the students. In cases where data is categorical or has repeated values, the mode can provide insights into trends that the mean and median may not reveal.

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