1. Problem statement: Create a lesson plan for the topic measures of dispersion that covers range, mean deviation, variance, and standard deviation and includes evaluation problems for ungrouped and grouped data. 2. Objective: Students will learn definitions, formulas, rules, worked examples, and practice calculations for both ungrouped and grouped data. 3. Key formulas and rules to show at the start: - Range formula and rule: $$\text{Range} = \max(x_i) - \min(x_i)$$ - Mean formula (arithmetic mean): $$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$ - Mean (mean absolute) deviation: $$\text{MD} = \frac{1}{n}\sum_{i=1}^{n} |x_i - \bar{x}|$$ - Population variance and standard deviation: $$\sigma^{2} = \frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^{2}$$ - Sample variance (Bessel correction) when appropriate: $$s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^{2}$$ - Standard deviation: $$\sigma = \sqrt{\sigma^{2}}\,.$$ 4. Important teaching rules and tips: - Explain conceptually that range measures spread by the extremes and that mean deviation, variance, and standard deviation measure average distance from the mean in different ways. - Emphasize that mean deviation uses absolute values so it is not algebraically convenient, while variance squares deviations so it is algebraically simpler. - Point out when to use population division by $n$ versus sample division by $n-1$. 5. Worked example 1 (Ungrouped data) — Problem 1: Compute range, mean, mean deviation, variance, and standard deviation for the data $[2,4,6,8,10,12]$. - Step a: Range calculation: $$\text{Range}=12-2=10$$ - Step b: Mean calculation: first compute the sum $\sum x_i=2+4+6+8+10+12=42$. - Show the mean formula and substitution: $$\bar{x}=\frac{\sum x_i}{n}=\frac{42}{6}$$ - Show intermediate cancellation when simplifying the fraction: $$\frac{42}{6}=\frac{\cancel{6}\cdot 7}{\cancel{6}}=7$$ - Step c: Mean deviation (MD): compute absolute deviations from mean $7$: the deviations are $|2-7|,|4-7|,|6-7|,|8-7|,|10-7|,|12-7|$ which give $5,3,1,1,3,5$. - Sum of absolute deviations is $5+3+1+1+3+5=18$ and then $$\text{MD}=\frac{18}{6}$$ - Show cancellation when simplifying: $$\frac{18}{6}=\frac{\cancel{6}\cdot 3}{\cancel{6}}=3$$ - Step d: Variance (population) calculations: squared deviations are $25,9,1,1,9,25$ and their sum is $70$. - Use the variance formula: $$\sigma^{2}=\frac{70}{6}$$ - Show intermediate simplification by factoring out common factor $2$: $$\frac{70}{6}=\frac{2\cdot 35}{2\cdot 3}=\frac{\cancel{2}\cdot 35}{\cancel{2}\cdot 3}=\frac{35}{3}\approx 11.6667$$ - Step e: Standard deviation: $$\sigma=\sqrt{\frac{35}{3}}\approx 3.41565$$ - Brief learner-friendly explanation for this example: the range is the spread between extremes, MD is the average absolute distance from the mean, variance averages squared distances (so units are squared), and standard deviation returns to original units by the square root. 6. Worked example 2 (Grouped data) — Problem 2: Evaluate measures for grouped data with midpoints and frequencies as follows: midpoints $x_i=5,10,15$ and frequencies $f_i=5,10,5$. - First compute total frequency $N=\sum f_i=5+10+5=20$. - Compute $\sum f_i x_i=5\cdot 5 + 10\cdot 10 + 5\cdot 15 = 200$. - Mean for grouped data using $\bar{x}=\dfrac{\sum f_i x_i}{\sum f_i}$: $$\bar{x}=\frac{200}{20}$$ - Show intermediate cancellation that simplifies neatly: $$\frac{200}{20}=\frac{\cancel{20}\cdot 10}{\cancel{20}}=10$$ - Compute mean absolute deviation for grouped data: weighted absolute deviations are $5\cdot |5-10| + 10\cdot |10-10| + 5\cdot |15-10| =5\cdot5 + 10\cdot0 +5\cdot5 =50$. - Mean deviation: $$\text{MD}=\frac{50}{20}$$ - Show cancellation when simplifying: $$\frac{50}{20}=\frac{\cancel{10}\cdot 5}{\cancel{10}\cdot 2}=\frac{5}{2}=2.5$$ - Compute variance for grouped data (population formula): weighted squared deviations are $5\cdot 25 + 10\cdot 0 + 5\cdot 25 =125+0+125=250$. - Variance: $$\sigma^{2}=\frac{250}{20}$$ - Show intermediate cancellation by factoring out 10: $$\frac{250}{20}=\frac{10\cdot 25}{10\cdot 2}=\frac{\cancel{10}\cdot 25}{\cancel{10}\cdot 2}=\frac{25}{2}=12.5$$ - Standard deviation: $$\sigma=\sqrt{\frac{25}{2}}\approx 3.53553$$ - Plain-language note: in grouped data we use class midpoints to approximate each class's contribution, and then treat the midpoints like observed values weighted by frequencies. 7. Classroom evaluation items (two distinct problems for practice and assessment): - Problem A (Ungrouped): Given data $[3,7,7,2,9,4,6]$, compute Range, Mean, Mean Deviation, Variance (use population formula), and Standard Deviation, and show all intermediate steps. - Problem B (Grouped): Given midpoints $2,6,10,14$ with frequencies $4,6,7,3$, compute the total frequency, mean, mean deviation, variance (population), and standard deviation, showing all intermediate steps and any cancellations used to simplify fractions. 8. Pedagogical sequence and timing suggestions: - Start with intuitive explanation and a quick range exercise (5 minutes). - Present formulas and a simple ungrouped worked example step-by-step (15 minutes). - Give a short guided practice on mean deviation (10 minutes). - Introduce variance and standard deviation with algebraic rationale and a worked ungrouped example (15 minutes). - Present grouped-data technique and the grouped worked example (15 minutes). - Finish with the two evaluation problems as independent practice or assessment (20 minutes). 9. Common student misconceptions and corrective tips: - Misconception: confusing mean deviation with variance signs; correct by emphasizing absolute values for MD and squaring for variance. - Misconception: forgetting to divide by $n$ or $n-1$ appropriately; correct by clarifying population vs sample contexts. - Tip: always compute mean first, then deviations, then square or absolute value as required, then average using the correct denominator. 10. Summary and final answer: The lesson plan includes definitions, formulas, rules, intermediate calculations with explicit cancellation steps where fractions are simplified, two evaluation problems (one ungrouped and one grouped), and suggested timings and pedagogical notes to teach measures of dispersion effectively.