Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate variance and standard deviation for ungrouped and grouped data.
• Interpret the meaning of variance and standard deviation.
• Compare the spread of different data sets.
• Apply these concepts to real-world scenarios.

Lesson summary

Statistics is a crucial branch of mathematics that helps us understand and interpret data. In South Africa, understanding statistics is vital for informed decision-making in various aspects of life, from understanding unemployment rates to analyzing crime statistics and making informed consumer choices. This week, we will focus on measures of dispersion: variance and standard deviation. These measures tell us how spread out the data is around the mean. A good understanding of these concepts is essential to making informed conclusions based on real world data.

Lesson notes

Measures of Dispersion: Variance and Standard Deviation While the mean, median, and mode tell us about the central tendency of a dataset, they don't tell us how spread out the data is. The variance and standard deviation are measures of dispersion that quantify this spread.

Variance: The variance measures the average squared deviation of each data point from the mean. A higher variance indicates that the data points are more spread out, while a lower variance suggests that the data points are clustered closer to the mean.

Standard Deviation: The standard deviation is the square root of the variance. It provides a measure of the spread of the data in the same units as the original data, making it easier to interpret. A smaller standard deviation implies the data points are closely clustered around the mean.

Formulas: Ungrouped Data: Variance (Population): σ² = Σ(xi - μ)² / N Where: σ² is the population variance xi is each individual data point μ is the population mean N is the total number of data points in the population Σ means 'sum of' Variance (Sample): s² = Σ(xi - x̄)² / (n - 1)

Where: s² is the sample variance xi is each individual data point x̄ is the sample mean n is the total number of data points in the sample (n-1) is called Bessel's correction, used to provide an unbiased estimate of population variance from sample data.

Standard Deviation (Population): σ = √σ² Standard Deviation (Sample): s = √s² Grouped Data: Variance (Population): σ² = Σ[f (xi - μ)²] / N Where: f is the frequency of each data point/interval xi is the midpoint of each interval μ is the population mean (calculated from the grouped data) N is the total number of data points (sum of frequencies)

Variance (Sample): s² = Σ[f (xi - x̄)²] / (n - 1)

Where: f is the frequency of each data point/interval xi is the midpoint of each interval x̄ is the sample mean (calculated from the grouped data) n is the total number of data points (sum of frequencies)

Standard Deviation (Population): σ = √σ² Standard Deviation (Sample): s = √s² Why (n-1) in Sample Variance? Using (n-1) instead of n in the sample variance formula (Bessel's correction) provides a better estimate of the population variance. If we used 'n', the sample variance would tend to underestimate the true population variance. This correction accounts for the fact that we are using the sample mean (x̄), which is itself an estimate, and thus introduces a degree of bias.

Worked example

Example 1: Ungrouped Data

The number of hours spent studying per week by 5 students are: 5, 7, 9, 11,

1

3. Calculate the sample variance and standard deviation.

Solution:

Calculate the sample mean (x̄):

x̄ = (5 + 7 + 9 + 11 + 13) / 5 = 45/5 = 9

Calculate the squared deviations from the mean:

(5 - 9)² = 16

(7 - 9)² = 4

(9 - 9)² = 0

(11 - 9)² = 4

(13 - 9)² = 16

Calculate the sum of squared deviations:

Σ(xi - x̄)² = 16 + 4 + 0 + 4 + 16 = 40

Calculate the sample variance (s²):

s² = Σ(xi - x̄)² / (n - 1) = 40 / (5 - 1) = 40 / 4 = 10

Calculate the sample standard deviation (s):

s = √s² = √10 ≈ 3.16

Interpretation: The standard deviation of 3.16 hours tells us that, on average, the hours spent studying deviate by approximately 3.16 hours from the mean of 9 hours.