Lesson Notes By Weeks and Term v5 - Grade 11
Statistics – Week 4 focus
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Statistics – Week 4 focus
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Subject: Mathematics
Class: Grade 11
Term: Term 4
Week: 4
Theme: General lesson support
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Statistics plays a crucial role in understanding and interpreting data around us, shaping our understanding of the world and informing decision-making. In South Africa, statistical literacy is essential for citizens to engage critically with information presented in the media, government reports, and research studies related to socio-economic issues, health trends, and environmental concerns. This week's focus is on measures of dispersion, particularly variance, standard deviation, and the interquartile range, and how these measures describe the spread or variability within a dataset.
Measures of Dispersion: These describe how spread out or clustered together the data points in a dataset are. They provide valuable information beyond measures of central tendency. The greater the dispersion, the more variability in the data.
Range: The simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset. While easy to calculate, it's highly susceptible to outliers.
Variance: A measure of how far each data point is from the mean. It's calculated as the average of the squared differences from the mean. Squaring the differences ensures that all values are positive and gives more weight to larger deviations.
Ungrouped Data Variance Formula: σ 2 = Σ(x i - μ) 2 / N (for a population) s 2 = Σ(x i - x̄) 2 / (n-1) (for a sample)
Where: σ 2 is the population variance s 2 is the sample variance x i is each individual data point μ is the population mean x̄ is the sample mean N is the population size n is the sample size Σ represents summation Grouped Data Variance Formula: s 2 = Σ[f i (x i - x̄) 2 ] / (n-1)
Where: s 2 is the sample variance f i is the frequency of each class interval x i is the midpoint of each class interval x̄ is the sample mean (calculated from grouped data) n is the total frequency (Σf i )
Standard Deviation: The square root of the variance. It is a more interpretable measure of dispersion because it's in the same units as the original data. A smaller standard deviation indicates that the data points are clustered closely around the mean, while a larger standard deviation indicates a wider spread.
Ungrouped Data Standard Deviation Formula: σ = √σ 2 (for a population) s = √s 2 (for a sample)
Grouped Data Standard Deviation Formula: s = √s 2 Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1). It represents the range of the middle 50% of the data. The IQR is less sensitive to outliers than the range or standard deviation.
Calculating Quartiles: To find Q1 and Q3, first, order the data from least to greatest. Q1 is the median of the lower half of the data. Q3 is the median of the upper half of the data.
Semi-Interquartile Range (SIQR): Half of the interquartile range. SIQR = (Q3 - Q1) / 2 The SIQR provides a measure of the average distance of the quartiles from the median, offering another perspective on data spread. Worked
Examples: Example 1: Ungrouped Data (Salaries of Fruit Vendors) The monthly salaries (in Rands) of 7 fruit vendors in a Johannesburg market are: 2500, 3000, 2800, 3200, 2700, 4000,
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0. Calculate the variance and standard deviation (treat as a sample).
Step 1: Calculate the sample mean (x̄): x̄ = (2500 + 3000 + 2800 + 3200 + 2700 + 4000 + 3500) / 7 = 2971.43 (approx.)
Step 2: Calculate the squared differences from the mean (x i - x̄) 2 : (2500-2971.43) 2 = 222222.45 (3000-2971.43) 2 = 815.73 (2800-2971.43) 2 = 29304.33 (3200-2971.43) 2 = 52272.45 (2700-2971.43) 2 = 73612.45 (4000-2971.43) 2 = 1058272.45 (3500-2971.43) 2 = 279304.33 Step 3: Sum the squared differences (Σ(x i - x̄) 2 ): Σ(x i - x̄) 2 = 1715404.19 Step 4: Calculate the sample variance (s 2 ): s 2 = 1715404.19 / (7-1) = 285900.70 (approx.)
Step 5: Calculate the sample standard deviation (s): s = √285900.70 = 534.69 (approx.)
Interpretation: The average salary of the fruit vendors is approximately R2971.43, and the spread of the salaries around this average is about R534.
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9. Example 2: Grouped Data (Ages of Library Users) The table below shows the ages of users at a local library: | Age Group | Frequency (f i ) | | :-------- | :--------------------- | | 10-20 | 15 | | 20-30 | 25 | | 30-40 | 30 | | 40-50 | 20 | | 50-60 | 10 | Calculate the variance and standard deviation (treat as a sample).
Step 1: Find the midpoint (x i ) of each class interval: 10-20: (10+20)/2 = 15 20-30: (20+30)/2 = 25 30-40: (30+40)/2 = 35 40-50: (40+50)/2 = 45 50-60: (50+60)/2 = 55 Step 2: Calculate the total frequency (n): n = 15 + 25 + 30 + 20 + 10 = 100 Step 3: Calculate the mean (x̄): x̄ = (1515 + 2525 + 3035 + 2045 + 10*55) / 100 = 34 Step 4: Calculate f i (x i - x̄) 2 for each class: 15(15-34) 2 = 5415 25(25-34) 2 = 2025 30(35-34) 2 = 30 20(45-34) 2 = 2420 10(55-34) 2 = 4410 Step 5: Sum the values from Step 4 (Σ[f i (x i - x̄) 2 ]): Σ[f i (x i - x̄) 2 ] = 5415 + 2025 + 30 + 2420 + 4410 = 14300 Step 6: Calculate the sample variance (s 2 ): s 2 = 14300 / (100-1) = 144.44 (approx.)
Step 7: Calculate the sample standard deviation (s): s = √144.44 = 12.02 (approx.)
Interpretation: The average age of the library users is approximately 34 years, and the spread of ages around this average is about 12.02 years.
Example 3: Interquartile Range (Test Scores) The scores of 10 students on a mathematics test are: 55, 62, 70, 75, 80, 82, 85, 90, 92,
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8. Calculate the interquartile range (IQR).
Step 1: Order the data: The data is already ordered.