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 and interpret Range and Interquartile Range (IQR).
• Calculate and interpret Variance and Standard Deviation.
• Compare the spread of different data sets.
• Understand the effect of outliers on measures of dispersion.

Lesson summary

This week, we delve deeper into the fascinating world of Statistics, building upon the foundational concepts you’ve already learned. Statistics isn't just about numbers; it's about understanding the stories those numbers tell. In South Africa, understanding statistics is vital for interpreting news reports about unemployment rates, analyzing crime statistics, and evaluating the impact of government policies. We need to be informed citizens who can critically assess data presented to us. This week, we focus specifically on measures of dispersion (range, interquartile range, variance, standard deviation) and how they help us to understand the spread and variability within a data set.

Lesson notes

Measures of Dispersion: Understanding Spread Measures of dispersion tell us how spread out or varied the data is in a set. A low dispersion means the data points are clustered closely around the mean, while a high dispersion means the data points are more scattered.

We'll explore four key measures: Range: Definition: The range is the simplest measure of dispersion. It is the difference between the highest and lowest values in a data set.

Formula: Range = Maximum value – Minimum value

Example: Consider the ages of students in a Grade 11 class: 15, 16, 16, 17, 17, 17,

1

8. The range is 18 – 15 = 3 years.

Interpretation: The ages of students in this class vary by a maximum of 3 years.

Limitations: The range is highly sensitive to outliers (extreme values).

Interquartile Range (IQR): Definition: The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1). It represents the range of the middle 50% of the data.

Formula: IQR = Q3 – Q1 Finding Quartiles: Arrange the data in ascending order.

Q2 (Median): The middle value. If there are an even number of data points, the median is the average of the two middle values. Q1: The median of the lower half of the data (excluding the median if the total number of data points is odd). Q3: The median of the upper half of the data (excluding the median if the total number of data points is odd).

Example: Consider the following test scores (out of 50): 20, 25, 28, 30, 32, 35, 38, 40, 42,

4

5. Ordered data: 20, 25, 28, 30, 32, 35, 38, 40, 42,

4

5. Q2 (Median): (32 + 35) / 2 = 33.5 Q1: (25 + 28) / 2 = 26.5 Q3: (40 + 42) / 2 = 41 IQR = 41 – 26.5 = 14.5 Interpretation: The middle 50% of test scores fall within a range of 14.5 marks.

Advantages: Less sensitive to outliers than the range.

Variance: Definition: Variance measures the average squared deviation of each data point from the mean. It quantifies the overall spread of the data around the mean. Formula (Population Variance, σ²): σ² = Σ(xᵢ - μ)² / N, where xᵢ is each data point, μ is the population mean, and N is the population size. Formula (Sample Variance, s²): s² = Σ(xᵢ - x̄)² / (n-1), where xᵢ is each data point, x̄ is the sample mean, and n is the sample size. Why (n-1) for sample variance? Using (n-1), called Bessel's correction, provides an unbiased estimate of the population variance when using a sample. This corrects for the fact that sample means tend to be closer to the data than the true population mean.

Example: Consider the number of hours a student spends studying each week: 2, 4, 6, 8, 10. (Assume this is a population) Mean (μ): (2 + 4 + 6 + 8 + 10) / 5 = 6 (xᵢ - μ)²: (2-6)² = 16; (4-6)² = 4; (6-6)² = 0; (8-6)² = 4; (10-6)² = 16 Σ(xᵢ - μ)²: 16 + 4 + 0 + 4 + 16 = 40 Variance (σ²): 40 / 5 = 8 Interpretation: The average squared deviation from the mean study time is 8 hours squared. While not intuitively meaningful due to the squaring, it's a crucial step towards calculating the standard deviation.

Standard Deviation: Definition: The standard deviation is the square root of the variance. It measures the typical distance of each data point from the mean, expressed in the same units as the original data. Formula (Population Standard Deviation, σ): σ = √(σ²) = √[Σ(xᵢ - μ)² / N] Formula (Sample Standard Deviation, s): s = √(s²) = √[Σ(xᵢ - x̄)² / (n-1)] Example (using the previous example): Variance (σ²) = 8 Standard Deviation (σ) = √8 ≈ 2.83 hours Interpretation: The study times typically deviate from the mean of 6 hours by approximately 2.83 hours. A lower standard deviation would suggest more consistent study habits across the group.

Importance: Standard deviation is widely used because it is easier to interpret than variance, being in the original unit of measurement.

Grouped Data: When dealing with grouped data (data presented in intervals or frequency tables), we need to use slightly modified formulas: Estimating the Mean: x̄ = Σ(fᵢ mᵢ) / Σfᵢ, where fᵢ is the frequency of each class interval and mᵢ is the midpoint of each class interval.

Estimating the Variance: s² = Σ[fᵢ (mᵢ - x̄)²] / (Σfᵢ - 1) (for sample data).

Estimating the Standard Deviation: s = √(s²)

Example (Grouped Data): The following table shows the heights (in cm) of Grade 11 learners in a class: | Height (cm) | Frequency (fᵢ) | |-------------|-----------------| | 150-155 | 5 | | 155-160 | 8 | | 160-165 | 12 | | 165-170 | 10 | | 170-175 | 5 | Calculate the midpoints (mᵢ): 152.5, 157.5, 162.5, 167.5, 172.5 Calculate the estimated mean (x̄): [(5 152.5) + (8 157.5) + (12 162.5) + (10 167.5) + (5 * 172.5)] / (5+8+12+10+5) = 163.25 cm Calculate Σ[fᵢ * (mᵢ - x̄)²]: This involves calculating (mᵢ - x̄)² for each interval, multiplying by the frequency, and summing the results. This yields approximately 1158.

7

5. Calculate the estimated variance (s²): 1158.75 / (40 - 1) ≈ 29.71 Calculate the estimated standard deviation (s): √29.71 ≈ 5.45 cm Therefore, the estimated standard deviation of the heights is approximately 5.45 cm.