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Statistics - Numerical Measures

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    Study Guide Statistics – Numerical Measures 1. Measures of Central Tendency (Mean, Median, Mode) When we collect a set of numbers, we often want to describe what a “typical” value looks like. Measures of central tendency help us do exactly that. These are numbers that tend to fall near the middle of a data set. The three most common measures are: Mean (average) Median (middle value) Mode (most frequent value) Example: Earnings for the Past Week Suppose your earnings for the week are shown in Table 1 : • Monday: $350 • Tuesday: $150 • Wednesday: $100 • Thursday: $350 • Friday: $50 1. Mean (The Average) The mean is what most people call the “average.” Definition The arithmetic mean is: Sum of all values ÷ Number of values Mean of the weekly earnings

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    Study Guide Add the earnings: 350 + 150 + 100 + 350 + 50 = 1000 Now divide by the number of days (5): 1000 ÷ 5 = 200 Mean = $200 So, if your earnings were spread out evenly across all five days, you would earn $200 per day . 2. Median (The Middle Value) The median is the value in the middle after the numbers are arranged in order . Step 1: Put the values in order $50, $100, $150, $350, $350 Step 2: Find the middle value The middle number is $150 . Median = $150

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    Study Guide What if there are an even number of values? If there is an even number of values, the median is the average of the two middle values . Example: 4, 10, 12, 26 The middle values are 10 and 12 (10 + 12) ÷ 2 = 11 Median = 11 Why the Median Can Be Better Than the Mean Sometimes a data set includes outliers , which are extreme values that are much higher or lower than the rest. When outliers exist, the mean can be misleading , but the median often gives a more “realistic” middle value. Example 1: Salaries in a Corporation Now look at the four salaries shown in Table 2 : • CEO: $1,000,000 • Manager: $50,000 • Administrative assistant: $30,000 • Custodian: $20,000 Mean Salary (1,000,000 + 50,000 + 30,000 + 20,000) ÷ 4 = 275,000 Mean = $275,000 Median Salary Order the salaries: $20,000, $30,000, $50,000, $1,000,000 There are 4 values, so take the middle two:

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    Study Guide $30,000 and $50,000 (30,000 + 50,000) ÷ 2 = 40,000 Median = $40,000 What does this show us? The CEO’s salary is a huge outlier , so the mean becomes much larger than the other salaries. In this case, the median ($40,000) is a better “typical salary” than the mean. 3. Mode (Most Common Value) The mode is the value that appears most often . In the weekly earnings: $50, $100, $150, $350, $350 Mode = $350 because it appears twice , and all the other values appear once. Notation and Formulas In statistics, the mean is written using special symbols. Sample Mean x̄

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    Study Guide Population Mean μ (pronounced “mew”) Sigma Symbol (Total/Sum) Σ This symbol means “add up all the values.” Formula for the Sample Mean This can also be written as: Where: • (n) = number of values in the data set Mean for Grouped Data Sometimes you don’t have every individual data value. Instead, the data is grouped into class intervals , like in a frequency table. Example: You might only know how many people fall into different salary ranges (like $25,000 – $29,999, etc.). In this case, you can still estimate the mean using midpoints . Formula for Mean of Grouped Data

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    Study Guide Where: • (x) = midpoint of the class interval • (f) = frequency (how many values are in that class) • (fx) = frequency × midpoint • (n) = total number of values Quick Example If: • midpoint (x = 8) • frequency (f = 10) Then: f x = 10(8) = 80 That means the interval contributes 80 to the total. Example: Garage Sale Prices (Grouped Data) Using Table 3 , we are given: • (n = 32) • ( Σ fx = 486) So the grouped - data mean is: Estimated mean price ≈ $15.19 Remember: this may not be the exact mean because we are using midpoints , not the real individual prices.

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    Study Guide Median for Grouped Data With grouped data, we may not know the exact middle value. But we can still get a good approximation . Step 1: Find the median position There are (n = 32) values. The median lies between the: • 16th value • 17th value From the frequency table, the median falls in the interval: $11.00 to $15.99

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    Study Guide Formula for Median of Grouped Data Where: • (L) = lower class limit of the median class • (n) = total number of measurements • (w) = class width • (f med ) = frequency of the median class • ( Σ f b ) = sum of frequencies before the median class Using the Garage Sale Data (Table 4) We already know the median class is $11.00 to $15.99 , so: • (L = 11) • (n = 32) • (w = 4.99) • (f med = 4) • ( Σ f b = 14) Substitute into the formula: Median ≈ 13.495

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    Study Guide As we already know, the median is located in class interval $11.00 to $15.99. So L = 11, n = 32, w = 4.99, f med = 4, and Σ f b = 14. Substituting into the formula:

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    Study Guide Symmetric Distributions A distribution is perfectly symmetric if the left side mirrors the right side. In a symmetric distribution: Mean = Median = Mode This is shown in Figure 1 , where all three measures fall at the same central point. Skewed Curves (Skewed Distributions) Sometimes a data set is not balanced evenly around the center. This often happens when there is an outlier , which is an unusually high or unusually low value. Why does this matter? An outlier can pull the mean (average) toward it, because the mean uses every value in the calculation. But the median usually stays near the center , because it depends only on the middle position — not on extreme values. When this happens, the graph of the data no longer looks perfectly symmetric. Instead, it becomes skewed , meaning it has a “tail” stretching more to one side.

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