Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Variance
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Variance
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Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 4
Theme: Statistics And Probarbilty
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Define the terms: mean, variance, coefficient of variance of the different probability distributions, Binomial, poisson and normal.
Mean (Expected Value, E(X) or $\mu$): The average or central tendency of a random variable. It represents the long-run average value of the outcome of a random experiment. For a discrete random variable, $E(X) = \sum x \cdot P(X=x)$. For a continuous random variable, $E(X) = \int x \cdot f(x) dx$. Variance ($\sigma^2$ or Var(X)): A measure of the spread or dispersion of a random variable around its mean. A small variance indicates that data points tend to be very close to the mean, while a large variance indicates that data points are spread out over a wider range.
Formula: Var(X) = $E[(X - \mu)^2]$ or, more commonly for calculations, Var(X) = $E(X^2) - (E(X))^2$. Standard Deviation ($\sigma$): The square root of the variance. It is expressed in the same units as the random variable, making it easier to interpret than variance. $\sigma = \sqrt{\text{Var(X)}}$.
Coefficient of Variation (CV): A standardized measure of dispersion that expresses the standard deviation as a percentage of the mean. It is useful for comparing the relative variability of two different datasets, even if they have different units or different means.
Formula: $CV = \frac{\sigma}{\mu} \times 100\%$ The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial.
Parameters: $n$: Number of trials. $p$: Probability of success in a single trial. $q$: Probability of failure, where $q = 1 - p$.
Probability Mass Function (PMF): $P(X=x) = \binom{n}{x} p^x q^{n-x}$, for $x = 0, 1, 2, ..., n$.
Mean of Binomial Distribution: Definition: The expected number of successes in $n$ trials.
Formula: $\mu = np$ Variance of Binomial Distribution: Definition: A measure of the spread of the number of successes around the mean.
Formula: $\sigma^2 = npq$ or $\sigma^2 = np(1-p)$ Standard Deviation of Binomial Distribution: Formula: $\sigma = \sqrt{npq}$ Coefficient of Variation of Binomial Distribution: Formula: $CV = \frac{\sqrt{npq}}{np} \times 100\% = \sqrt{\frac{q}{np}} \times 100\%$ Worked Example 1 (Binomial): A quality control inspector at a Nigerian beverage company finds that 15% of bottled drinks have defects. If a random sample of 20 bottles is selected, calculate the mean, variance, and coefficient of variation of the number of defective bottles.
Solution: Given: Number of trials, $n = 20$ Probability of success (a bottle being defective), $p = 0.15$ Probability of failure (a bottle not being defective), $q = 1 - p = 1 - 0.15 = 0.85$ Mean ($\mu$): $\mu = np = 20 \times 0.15 = 3$ Interpretation: On average, 3 out of 20 bottles are expected to be defective. Variance ($\sigma^2$): $\sigma^2 = npq = 20 \times 0.15 \times 0.85 = 2.55$ Standard Deviation ($\sigma$): $\sigma = \sqrt{2.55} \approx 1.597$ Coefficient of Variation (CV): $CV = \frac{\sigma}{\mu} \times 100\% = \frac{1.597}{3} \times 100\% \approx 53.23\%$ Interpretation: The standard deviation is approximately 53.23% of the mean. This indicates a moderate level of variability in the number of defective bottles. The Poisson distribution describes the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence and that these events occur independently. It is often used for rare events.
Parameter: $\lambda$ (lambda): The average rate of occurrence of the event in the given interval (mean number of events).
Probability Mass Function (PMF): $P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!}$, for $x = 0, 1, 2, ...$ Mean of Poisson Distribution: Definition: The average number of events in the given interval.
Formula: $\mu = \lambda$ Variance of Poisson Distribution: Definition: A measure of the spread of the number of events around the average. A unique property of the Poisson distribution is that its mean and variance are equal.
Formula: $\sigma^2 = \lambda$ Standard Deviation of Poisson Distribution: Formula: $\sigma = \sqrt{\lambda}$ Coefficient of Variation of Poisson Distribution: Formula: $CV = \frac{\sqrt{\lambda}}{\lambda} \times 100\% = \frac{1}{\sqrt{\lambda}} \times 100\%$ Worked Example 2 (Poisson): The average number of power outages in a particular Nigerian community during the rainy season is 3 per week. Assuming the number of outages follows a Poisson distribution, calculate the mean, variance, and coefficient of variation.
Solution: Given: Average rate of occurrence, $\lambda = 3$ events per week. Mean ($\mu$): $\mu = \lambda = 3$ Interpretation: On average, there are 3 power outages per week. Variance ($\sigma^2$): $\sigma^2 = \lambda = 3$ Standard Deviation ($\sigma$): $\sigma = \sqrt{3} \approx 1.732$ Coefficient of Variation (CV): $CV = \frac{\sigma}{\mu} \times 100\% = \frac{1.732}{3} \times 100\% \approx 57.73\%$ Interpretation: The standard deviation is about 57.73% of the mean. This indicates a significant level of week-to-week variability in the number of power outages. The Normal distribution (or Gaussian distribution) is a continuous probability distribution that is symmetric around its mean, forming a bell-shaped curve. It is one of the most important distributions in statistics due to the Central Limit Theorem and its widespread occurrence in natural phenomena and empirical data.
Parameters: $\mu$ (mu): The mean of the distribution, which also represents the median and mode. $\sigma^2$ (sigma squared): The variance of the distribution.
Probability Density Function (PDF): $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$, for $-\infty < x < \infty$.
Note for SS3:* Students are generally not expected to work with the PDF directly for calculations of mean and variance, but rather to identify the parameters from the definition of a Normal distribution $X \sim N(\mu, \sigma^2)$.
Mean of Normal Distribution: Definition: The central value of the distribution, where the peak of the bell curve lies.
Formula: $\mu$ (It is one of the defining parameters)
Variance of Normal Distribution: Definition: A measure of how spread out the bell curve is. A larger variance results in a wider, flatter curve.
Formula: $\sigma^2$ (It is one of the defining parameters)
Standard Deviation of Normal Distribution: Formula: $\sigma$ Coefficient of Variation of Normal Distribution: Formula: $CV = \frac{\sigma}{\mu} \times 100\%$ Worked Example 3 (Normal): The scores of SS3 students in a national Further Mathematics examination are normally distributed with a mean of 65 marks and a variance of
1
0
0. Calculate the mean, variance, and coefficient of variation of these scores.
Solution: Given: Mean, $\mu = 65$ marks Variance, $\sigma^2 = 100$ Mean ($\mu$): $\mu = 65$ marks Interpretation: The average score for SS3 students in this examination is 65 marks. Variance ($\sigma^2$): $\sigma^2 = 100$ Standard Deviation ($\sigma$): $\sigma = \sqrt{100} = 10$ marks Coefficient of Variation (CV): $CV = \frac{\sigma}{\mu} \times 100\% = \frac{10}{65} \times 100\% \approx 15.38\%$ Interpretation: The standard deviation of scores is approximately 15.38% of the mean score. This indicates a relatively low variability in student performance, suggesting that most students scored fairly close to the average.
Agriculture and Food Security: Application: Farmers in Nigeria can use variance to assess the consistency of crop yields (e.g., maize, rice) under different farming techniques, soil types, or fertilizer applications. A low variance in yield for a particular method indicates reliability, making it a preferable choice for ensuring food security.
Context: Comparing the variability of yam yields from farms using traditional methods versus modern irrigation systems in Benue State. A lower coefficient of variation would imply more stable and predictable harvests, which is vital for farmer income and national food supply.
Public Health and Disease Control: Application: Public health officials can analyze the variance in the number of reported cases of diseases like malaria or cholera across different weeks or communities. A sudden increase in variance could signal an outbreak or an inconsistency in prevention efforts, prompting targeted interventions.
Context: Monitoring the variance in the weekly count of cholera cases in densely populated areas of Lagos. High variance suggests unpredictable spikes, requiring a more dynamic response strategy, whereas a low variance might indicate a stable (though not necessarily desirable) baseline. Business, Finance, and Quality Control: Application: Nigerian manufacturing companies use variance in quality control to ensure product consistency (e.g., the weight of cement bags, volume of bottled water, or durability of tyres). High variance indicates production issues, leading to product recalls or customer dissatisfaction. In finance, variance (or standard deviation) measures the risk of an investment (e.g., the volatility of shares on the Nigerian Stock Exchange).
Context: A cement factory in Ogun State monitors the variance of the weight of its 50kg bags. If the variance is too high, it means some bags are significantly underweight (leading to customer complaints) and others are overweight (leading to financial loss for the company). Investors use the coefficient of variation to compare the risk-return profiles of different Nigerian companies' stocks, choosing investments with lower relative risk.