Describing Data: Measures of Center and Average

Introduction to Describing Data: What is Average?

• Learning Goals for Section 4.1:

  • Understand the fundamental differences between the three primary measures of center: mean, median, and mode.

  • Analyze how outliers affect these different measures of average.

  • Determine when it is appropriate to apply a weighted mean instead of a standard mean.

Measures of Center: Mean, Median, and Mode

• The Mean:

  • Definition: The mean is the value most commonly referred to as the "average."

  • Calculation: It is calculated by summing all the values in a data set and then dividing by the total number of values.

  • Metaphor/Visualization (Figure 4.1): If a histogram were constructed using physical blocks, the mean would represent the specific point on the horizontal axis where the distribution would balance perfectly.

• The Median:

  • Definition: The median is the value that occupies the middle position when the data set is sorted in ascending or descending order.

  • Calculating for Even Data Sets: If the data set contains an even number of values, the median is defined as the value halfway between the two middle values (calculated as the average of those two values).

• The Mode:

  • Definition: The mode is the most frequently occurring value (or group of values) in a data set.

Rounding Rule for Statistical Calculations

• General Principle: When performing statistical calculations, the final answer should typically be expressed with one more decimal place of precision than what is provided in the original list of data values.

• Examples:

  • If original data are whole numbers (0 decimal places), the mean should be rounded to the nearest tenth ($0.1$).

  • If original data are given to the nearest tenth (1 decimal place), the mean should be rounded to the nearest hundredth ($0.01$).

• Precision Note: One must always round only the final answer; do not round any intermediate values used during the calculation process.

Practical Application: Price Data Example

• DataSet: Eight grocery stores sell a PR energy bar at the following eight prices:

  • $\$1.79$

  • $\$1.29$

  • $\$1.29$

  • $\$1.35$

  • $\$1.39$

  • $\$1.49$

  • $\$1.59$

  • $\$1.09$

• Calculating the Mean:

  • Sum of prices: $\$1.79 + \$1.29 + \$1.29 + \$1.35 + \$1.39 + \$1.49 + \$1.59 + \$1.09 = \$11.28$

  • Count ($n$): $8$

  • Mean = $\frac{\$11.28}{8} = \$1.41$

  • Using the Rounding Rule ($3$ decimal places): $\$1.410$

• Calculating the Median:

  • Sorted Data (Ascending Order): $\$1.09, \$1.29, \$1.29, \$1.35, \$1.39, \$1.49, \$1.59, \$1.79$

  • Since there are $8$ values, the middle consists of the fourth and fifth values: $\$1.35$ and $\$1.39$.

  • Calculation: $\frac{\$1.35 + \$1.39}{2} = \$1.37$

  • Using the Rounding Rule ($3$ decimal places): $\$1.370$

• Calculating the Mode:

  • The mode is $\$1.29$ because it appears twice, which is more frequent than any other price in the set.

The Impact of Outliers on Statistical Measures

• Definition of Outlier: An outlier (or outlying value) is a value in a data set that is significantly higher or significantly lower than almost all other values.

• The Basketball Contract Scenario:

  • Five graduating seniors receive first-year contract offers for the NBA. Four receive no offer ($0$), and one receives $\$10,000,000$.

  • Data: $0, 0, 0, 0, \$10,000,000$

  • Mean Offer Calculation: $\frac{\$0 + \$0 + \$0 + \$0 + \$10,000,000}{5} = \$2,000,000$

  • Problem of Representation: While the mean indicates an "average" of $\$2 million$, this number is unrepresentative because $\$10,000,000$ is an extreme outlier. If the outlier is removed, the mean drops to zero.

• Resistance to Outliers:

  • Mean: Significantly affected by outliers; they pull the mean toward the extreme value.

  • Median: Generally unaffected because outliers exist at the ends of the sorted list, not the center. (Note: Deleting an outlier may change the count of values and thus shift the median slightly, but the value of the outlier itself does not distort the result).

  • Mode: Generally unaffected by outliers.

"Average" Confusion and Real-World Examples

• The Wage Dispute Scenario:

  • Context: News reports an average wage of $\$42$\,per hour in the industry. Workers at a firm claim their average is only $\$36$, while management claims the firm's average is $\$48$.

  • Resolution: Both can be correct if they use different measures of center.

  • Hypothetical Data: Five workers with wages $\$36, \$36, \$36, \$36,$ and $\$960$.

  • Median calculation: The middle value is $\$36$.

  • Mean calculation: According to the text, the mean of such a distribution is reported as $\$48$.

• Confusion Source: Misunderstandings often arise when the specific type of "average" (mean vs. median) is not specified, or when calculating methods are not transparent.

Calculating Weighted Mean

• Definition: A weighted mean accounts for variations in the relative importance or "weight" of individual data values within a set.

• Formula: $\text{Weighted Mean} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}}$

• Course Grade Example:

  • Structure: 4 Quizzes (each worth $15\%$) and 1 Final Exam (worth $40\%$).

  • Scores: Quiz scores are $75, 80, 84, 88$; Final Exam score is $96$.

  • Calculation using percentages as weighted values (15 and 40):

    • Sum of weighted values: $(75 \times 15) + (80 \times 15) + (84 \times 15) + (88 \times 15) + (96 \times 40) = 1125 + 1200 + 1260 + 1320 + 3840 = 8745$

    • Sum of weights: $15 + 15 + 15 + 15 + 40 = 100$

    • Final Score: $\frac{8745}{100} = 87.45$

  • Rounding Rule Application: The finale score is rounded to $87.5$.

Questions & Discussion

• Think About It (Weights as Decimals):

  • Question: Because weights often represent percentages, would calculating the weighted mean using decimals (e.g., $0.15$ and $0.40$) change the final answer?

  • Logic: No, the answer remains the same because both the numerator ($\sum (x \times w)$) and the denominator ($\sum w$) are scaled by the same factor, maintaining the same ratio.

Formalizing the Mean with Summation Notation

• The Summation Sign (\Sigma): This Greek capital letter sigma indicates that a set of numbers should be added together.

• Variables:

  • $x$: Represents each individual value in a data set.

  • $n$: Represents the total number of values in a sample.

  • $\bar{x}$: The standard symbol for the mean of a sample.

  • $\mu$ (mu): The Greek letter used to represent the mean of a population.

• General Formulas:

  • Mean: $\bar{x} = \frac{\sum x}{n}$

  • Weighted Mean: $\text{Weighted Mean} = \frac{\sum (x \times w)}{\sum w}$

Calculating Measures for Binned Data

• Approach: For data organized into bins (ranges), assume that the middle value of the bin represents every data value within that bin.

• Binned Data Example (Table of 50 values):

  • Bin 1 (0–6): Middle value = $3$. Frequency = $10$. Contribution = $(3 \times 10) = 30$.

  • Bin 2 (7–13): Middle value = $10$. Frequency = $12$. Contribution = $(10 \times 12) = 120$.

  • Bin 3 (14–20): Middle value = $17$. Frequency = $11$. Contribution = $(17 \times 11) = 187$.

  • Bin 4 (21–27): Middle value = $24$. Frequency = $17$. Contribution = $(24 \times 17) = 408$.

• Calculating the Mean:

  • Total Sum: $30 + 120 + 187 + 408 = 745$

  • Total Count ($n$): $50$

  • Mean: $\frac{745}{50} = 14.9\,\text{years}$

• Determining the Median and Mode:

  • Median: With $50$ values, the median is located between the $25\text{th}$ and $26\text{th}$ sorted values. Counting frequencies ($10+12=22$), the $25\text{th}$ and $26\text{th}$ values fall into the $14\text{--}20$ bin, known as the median class.

  • Mode: The bin with the highest frequency. In this data set, the mode is the $21\text{--}27$ bin (frequency of $17$).