Lesson Notes By Weeks and Term v5 - Grade 11
Statistics – Week 3 focus
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Statistics – Week 3 focus
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Subject: Mathematics
Class: Grade 11
Term: Term 4
Week: 3
Theme: General lesson support
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Statistics is the science of collecting, analysing, interpreting, and presenting data. In Grade 11, we expand on the statistical concepts learned in previous grades, focusing on measures of dispersion, skewness and data representation techniques that allow us to draw meaningful conclusions about populations based on samples. This is crucial in understanding trends, making informed decisions, and solving problems in various aspects of life, from analysing crime rates in our communities to predicting the outcome of elections or assessing the success of interventions designed to address social issues.
2.1 Measures of Dispersion: Variance and Standard Deviation Measures of dispersion describe the spread or variability of data values around the mean. A small dispersion indicates that the data points are clustered closely around the mean, while a large dispersion suggests that the data points are more spread out.
Variance: The variance is a measure of how spread out the data is from the mean. It is calculated as the average of the squared differences from the mean.
Formula for Ungrouped Data: σ² = Σ(xᵢ - μ)² / N, where σ² is the population variance, xᵢ is each data point, μ is the population mean, and N is the total number of data points. For a sample, use s² = Σ(xᵢ - x̄)² / (n-1), where s² is the sample variance, x̄ is the sample mean, and n is the sample size. We use (n-1) for the sample variance to provide an unbiased estimate of the population variance.
Formula for Grouped Data: σ² = Σfᵢ(xᵢ - μ)² / Σfᵢ, where fᵢ is the frequency of each data point xᵢ. For a sample, use s² = Σfᵢ(xᵢ - x̄)² / (Σfᵢ - 1).
Standard Deviation: The standard deviation is the square root of the variance. It measures the average distance of the data points from the mean. It is in the same units as the original data, making it easier to interpret than the variance.
Formula for Ungrouped Data: σ = √σ² (population) or s = √s² (sample).
Formula for Grouped Data: σ = √σ² (population) or s = √s² (sample).
Example 1: Ungrouped Data (Salaries of Employees) Suppose we have the monthly salaries (in Rands) of 5 employees in a small business: 8000, 10000, 12000, 14000,
1
6
0
0
0. Calculate the variance and standard deviation (treating this as a sample).
Calculate the Mean: x̄ = (8000 + 10000 + 12000 + 14000 + 16000) / 5 = 12000 Calculate the squared differences from the mean: (8000 - 12000)² = 16000000 (10000 - 12000)² = 4000000 (12000 - 12000)² = 0 (14000 - 12000)² = 4000000 (16000 - 12000)² = 16000000 Calculate the Sample Variance (s²): s² = (16000000 + 4000000 + 0 + 4000000 + 16000000) / (5-1) = 40000000 / 4 = 10000000 Calculate the Sample Standard Deviation (s): s = √10000000 ≈ 3162.28 Interpretation: The average monthly salary is R12000, and the standard deviation is approximately R3162.
2
8. This means that the salaries typically deviate from the mean by about R3162.
2
8. Example 2: Grouped Data (Ages of Students in a Class) The following table shows the ages of students in a class: | Age (Years) | Frequency | | :---------- | :-------- | | 16 | 5 | | 17 | 10 | | 18 | 8 | | 19 | 2 | Calculate the variance and standard deviation (treating this as the entire population). Calculate the Mean (μ): μ = (16\5 + 17\10 + 18\8 + 19\2) / (5 + 10 + 8 + 2) = (80 + 170 + 144 + 38) / 25 = 432 / 25 = 17.28 Calculate the squared differences from the mean and multiply by the frequency: 16: (16 - 17.28)² \ 5 = 7.84 \* 5 = 39.2 17: (17 - 17.28)² \ 10 = 0.0784 \* 10 = 0.784 18: (18 - 17.28)² \ 8 = 0.5184 \* 8 = 4.1472 19: (19 - 17.28)² \ 2 = 2.9584 \* 2 = 5.9168 Calculate the Population Variance (σ²): σ² = (39.2 + 0.784 + 4.1472 + 5.9168) / 25 = 50.048 / 25 = 2.00192 Calculate the Population Standard Deviation (σ): σ = √2.00192 ≈ 1.41 Interpretation: The average age is 17.28 years, and the standard deviation is approximately 1.41 years. 2.2 Interquartile Range (IQR) and Semi-Interquartile Range Interquartile Range (IQR): The IQR is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q3 – Q
1. It describes the spread of the middle 50% of the data.
Semi-Interquartile Range: The semi-interquartile range is half of the interquartile range, (Q3 – Q1)/
2. It represents half the distance between the first and third quartiles.
Example 3: Finding the IQR Given the following data set of test scores: 50, 60, 65, 70, 75, 80, 85, 90, 95, 100 Order the Data: The data is already ordered.
Find Q1 (25th percentile): Q1 is the median of the lower half of the data.
Lower half: 50, 60, 65, 70,
7
5. The median is
6
5. Therefore, Q1 =
6
5. Find Q3 (75th percentile): Q3 is the median of the upper half of the data.
Upper half: 80, 85, 90, 95,
1
0
0. The median is
9
0. Therefore, Q3 =
9
0. Calculate the IQR: IQR = Q3 - Q1 = 90 - 65 = 25 The IQR is 25. 2.3 Skewness and Data Representation Skewness refers to the asymmetry of a distribution. It indicates whether the data is concentrated on one side of the mean or evenly distributed.
Symmetrical Distribution: The data is evenly distributed around the mean. The mean, median, and mode are approximately equal.
Positively Skewed (Right Skewed): The tail is longer on the right side. The mean is greater than the median.
Negatively Skewed (Left Skewed): The tail is longer on the left side. The mean is less than the median.
Visual Representation: Box-and-Whisker Plots: A box-and-whisker plot visually represents the median, quartiles (Q1 and Q3), and minimum and maximum values of a data set.