Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Calculate variance and standard deviation for ungrouped data.
• Determine and use the interquartile range (IQR).
• Interpret the meaning of these measures in real-world scenarios.
• Compare the spread of two or more datasets.
• Explain the relationship between variance and standard deviation.

Lesson summary

This week, we delve deeper into the fascinating world of Statistics, building upon the foundational knowledge gained in previous weeks. Specifically, we'll be focusing on measures of dispersion: variance, standard deviation, and the interquartile range (IQR). Understanding these measures is crucial because they allow us to quantify the spread or variability within a data set. In a South African context, this is particularly important for analyzing socioeconomic data, understanding income inequality, assessing the effectiveness of educational interventions, and even predicting agricultural yields based on rainfall patterns.

Lesson notes

2.1 Measures of Dispersion: Introduction Measures of dispersion describe how spread out or clustered the data values are around the central tendency (mean, median, mode). A small dispersion indicates that the data points are clustered closely around the mean, while a large dispersion indicates a wider spread. 2.2 Variance The variance measures the average squared deviation of each data point from the mean. Squaring the deviations ensures that both positive and negative deviations contribute to the overall spread. Formula for Population Variance (σ²): σ² = Σ(xᵢ - μ)² / N Where: σ² = Population variance xᵢ = Each individual data point μ = Population mean N = Total number of data points in the population Σ = Summation Formula for Sample Variance (s²): s² = Σ(xᵢ - x̄)² / (n - 1)

Where: s² = Sample variance xᵢ = Each individual data point in the sample x̄ = Sample mean n = Total number of data points in the sample Σ = Summation Why do we use (n-1) for the sample variance? Using (n-1), known as Bessel's correction, provides a more unbiased estimate of the population variance when using sample data. This accounts for the fact that sample means tend to underestimate population means. 2.3 Standard Deviation The standard deviation is the square root of the variance. It provides a measure of spread in the same units as the original data, making it easier to interpret than the variance. Formula for Population Standard Deviation (σ): σ = √(σ²) = √[Σ(xᵢ - μ)² / N] Formula for Sample Standard Deviation (s): s = √(s²) = √[Σ(xᵢ - x̄)² / (n - 1)] Why is standard deviation important? Because its units are the same as the original data, making it directly comparable to the average value. 2.4 Interquartile Range (IQR) The interquartile range (IQR) measures the spread of the middle 50% of the data. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). IQR = Q3 - Q1 How to find Q1 and Q3: Arrange the data in ascending order. Find the median (Q2). Q1 is the median of the lower half of the data (excluding Q2 if it's part of the data). Q3 is the median of the upper half of the data (excluding Q2 if it's part of the data). The IQR is resistant to outliers, meaning that extreme values do not significantly affect its value. This makes it a useful measure of spread when the data contains outliers. 2.5 Worked Examples Example 1: Calculating Variance and Standard Deviation Consider the following data set representing the number of learners absent from a Grade 11 class over 5 days: 2, 3, 4, 5,

6. Calculate the mean (x̄): x̄ = (2 + 3 + 4 + 5 + 6) / 5 = 4 Calculate the deviations from the mean (xᵢ - x̄): 2 - 4 = -2 3 - 4 = -1 4 - 4 = 0 5 - 4 = 1 6 - 4 = 2 Square the deviations (xᵢ - x̄)²: (-2)² = 4 (-1)² = 1 (0)² = 0 (1)² = 1 (2)² = 4 Sum the squared deviations: Σ(xᵢ - x̄)² = 4 + 1 + 0 + 1 + 4 = 10 Calculate the sample variance (s²): s² = 10 / (5 - 1) = 10 / 4 = 2.5 Calculate the sample standard deviation (s): s = √2.5 ≈ 1.58 Interpretation: The average number of learners absent is 4, and the data points typically deviate from the mean by approximately 1.58 learners.

Example 2: Calculating the IQR Consider the following data set representing the ages of members in a stokvel: 25, 30, 35, 40, 45, 50, 55, 60,

6

5. Arrange the data in ascending order: (Already done)

Find the median (Q2): The median is

4

5. Find Q1: The lower half of the data is 25, 30, 35,

4

0. The median of this lower half is (30 + 35)/2 = 32.

5. Therefore, Q1 = 32.

5. Find Q3: The upper half of the data is 50, 55, 60,

6

5. The median of this upper half is (55 + 60)/2 = 57.

5. Therefore, Q3 = 57.

5. Calculate the IQR: IQR = Q3 - Q1 = 57.5 - 32.5 =

2

5. Interpretation: The middle 50% of the ages in the stokvel range from 32.5 to 57.5, with a spread of 25 years.

Example 3: Comparing Data Sets Two different schools, School A and School B, administered the same Math test.

Here are the results (out of 100): School A: Mean = 65, Standard Deviation = 10 School B: Mean = 70, Standard Deviation = 5 Analysis: While School B has a higher average score, School A has a larger standard deviation. This suggests that the scores at School A are more spread out than those at School B. In School B, the learners' performance is more consistent. Guided Practice (With Solutions)

Question 1: Calculate the variance and standard deviation for the following set of data representing the daily wages (in Rands) of 5 farm workers: 80, 90, 100, 110, 120.