Statistics - Numerical Measures - Quick Guides

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Study GuideStatistics–Numerical Measures1.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 tendencyhelp us do exactly that.These are numbers that tend to fall near themiddleof a data set. The three most common measuresare:Mean(average)Median(middle value)Mode(most frequent value)Example: Earnings for the Past WeekSuppose your earnings for the week are shown inTable 1:•Monday: $350•Tuesday: $150•Wednesday: $100•Thursday: $350•Friday: $501. Mean (The Average)Themeanis what most people call the “average.”DefinitionThearithmetic meanis:Sum of all values ÷ Number of valuesMean of the weekly earnings

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

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Study GuideWhat if there are an even number of values?If there is anevennumber of values, the median is theaverage of the two middle values.Example: 4, 10, 12, 26The middle values are10 and 12(10 + 12) ÷ 2 = 11Median = 11Why the Median Can Be Better Than the MeanSometimes a data set includesoutliers, which are extreme values that are much higher or lower thanthe rest.When outliers exist, themean can be misleading, but themedian often gives a more “realistic”middle value.Example 1: Salaries in a CorporationNow look at the four salaries shown inTable 2:•CEO: $1,000,000•Manager: $50,000•Administrative assistant: $30,000•Custodian: $20,000Mean Salary(1,000,000 + 50,000 + 30,000 + 20,000) ÷ 4 = 275,000Mean = $275,000Median SalaryOrder the salaries:$20,000, $30,000, $50,000, $1,000,000There are 4 values, so take the middle two:

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Study Guide$30,000 and $50,000(30,000 + 50,000)÷2 = 40,000Median = $40,000What does this show us?TheCEO’s salary is a huge outlier, so the mean becomesmuch largerthan the other salaries.In this case, themedian ($40,000)is a better “typical salary” than the mean.3. Mode (Most Common Value)Themodeis the value that appearsmost often.In the weekly earnings:$50, $100, $150, $350, $350Mode = $350because it appearstwice, and all the other values appear once.Notation and FormulasIn statistics, the mean is written using special symbols.Sample Meanx̄

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Study GuidePopulation Meanμ(pronounced “mew”)Sigma Symbol (Total/Sum)ΣThis symbol means “add up all the values.”Formula for the Sample MeanThis can also be written as:Where:•(n) = number of values in the data setMean for Grouped DataSometimes you don’t have every individual data value. Instead, the data is grouped intoclassintervals, 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 usingmidpoints.Formula for Mean of Grouped Data

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Study GuideWhere:•(x) = midpoint of the class interval•(f) = frequency (how many values are in that class)•(fx) = frequency × midpoint•(n) = total number of valuesQuick ExampleIf:•midpoint (x = 8)•frequency (f = 10)Then:fx = 10(8) = 80That means the interval contributes80to the total.Example: Garage Sale Prices (Grouped Data)UsingTable 3, we are given:•(n = 32)•(Σfx = 486)So the grouped-data mean is:Estimated mean price ≈ $15.19Remember: this may not be the exact mean because we are usingmidpoints, not the real individualprices.

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Study GuideMedian for Grouped DataWith grouped data, we may not know the exact middle value. But we can still get agoodapproximation.Step 1: Find the median positionThere are (n = 32) values.The median lies between the:•16th value•17th valueFrom the frequency table, the median falls in the interval:$11.00 to $15.99

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Study GuideFormula for Median of Grouped DataWhere:•(L) = lower class limit of the median class•(n) = total number of measurements•(w) = class width•(fmed) = frequency of the median class•(Σfb) = sum of frequencies before the median classUsing 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)•(fmed= 4)•(Σfb= 14)Substitute into the formula:Median ≈ 13.495

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Study GuideAs we already know, the median is located in class interval $11.00 to $15.99. SoL= 11,n= 32,w=4.99,fmed= 4, andΣfb= 14.Substituting into the formula:

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Study GuideSymmetric DistributionsA distribution isperfectly symmetricif the left side mirrors the right side.In a symmetric distribution:Mean = Median = ModeThis is shown inFigure 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 anoutlier, which is an unusually high or unusually low value.Why does this matter?Anoutlier can pull the mean(average) toward it, because the mean uses every value in thecalculation.But themedian usually stays near the center, because it depends only on the middle position—noton extreme values.When this happens, the graph of the data no longer looks perfectly symmetric. Instead, it becomesskewed, meaning it has a “tail” stretching more to one side.

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