Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Correlation
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Correlation
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Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 3
Theme: Statistics And Probarbilty
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Know the meaning of correlation: negative, zero and positive correlation from scatter diagrams, rank correlation, spearmans rank and correlation coefficient Use data with out ties
80 | | 2 | 7 | 150 | | 3 | 2 | 60 | | 4 | 10 | 180 | | 5 | 5 | 120 | Calculate Spearman's Rank Correlation Coefficient between years of experience and monthly sales.
Solution 3:
1. Rank X (Years of Experience): (Highest experience gets rank 1) 10 years (Rank 1) 7 years (Rank 2) 5 years (Rank 3) 3 years (Rank 4) 2 years (Rank 5)
2. Rank Y (Monthly Sales): (Highest sales gets rank 1) ₦180,000 (Rank 1) ₦150,000 (Rank 2) ₦120,000 (Rank 3) ₦80,000 (Rank 4) ₦60,000 (Rank 5)
3. Number of pairs (n): $n = 5$.
4. Calculate d, d2, and $\sum d^2$: | Employee | X (Experience) | Y (Sales) | Rank X | Rank Y | d (Rank X - Rank Y) | d2 | | :------- | :------------- | :-------- | :----- | :----- | :------------------ | :-- | | 1 | 3 | 80 | 4 | 4 | 0 | 0 | | 2 | 7 | 150 | 2 | 2 | 0 | 0 | | 3 | 2 | 60 | 5 | 5 | 0 | 0 | | 4 | 10 | 180 | 1 | 1 | 0 | 0 | | 5 | 5 | 120 | 3 | 3 | 0 | 0 | | Total| | | | | 0 | 0 | $\sum d^2 = 0$.
5. Apply the formula: $$ \rho = 1 - \frac{6 \sum d^2}{n(n^2 - 1)} $$ $$ \rho = 1 - \frac{6 \times 0}{5(5^2 - 1)} $$ $$ \rho = 1 - \frac{0}{120} $$ $$ \rho = 1 - 0 $$ $$ \rho = 1 $$
Commentary: The Spearman's rank correlation coefficient is $1$. This indicates a perfect positive rank correlation. It means that the employees' ranks in terms of years of experience perfectly match their ranks in terms of monthly sales. The employee with the highest experience has the highest sales, the second highest experience has the second highest sales, and so on. The teacher should guide students through these questions, providing support and correcting errors as they arise. Question 1 (Interpretation from Scatter Diagram): A local government official plotted the number of boreholes drilled in different communities against the average reported cases of waterborne diseases in those communities over a year. The scatter diagram showed points generally moving downwards from left to right. a) What type of correlation does this suggest between the number of boreholes and waterborne disease cases? b) Is this a strong or weak correlation, given the general trend is clear but some points are a bit scattered? c) What does this correlation imply for public health intervention?
Solution 1: a) This suggests a negative correlation. As the number of boreholes (presumably improving access to clean water) increases, the cases of waterborne diseases tend to decrease. b) If the general trend is clear but points are a bit scattered, it suggests a moderate to weak negative correlation. c) This correlation implies that increasing the number of boreholes could be an effective public health intervention to reduce waterborne diseases, although other factors might also be at play. Question 2 (Calculating Spearman's Rank Correlation Coefficient - Data Without Ties): A panel of judges ranked 5 participants in a local talent show (ranging from 1st to 5th place) based on their singing ability (Judge A) and dancing ability (Judge B). | Participant | Judge A (Singing Rank) | Judge B (Dancing Rank) | | :---------- | :--------------------- | :--------------------- | | P | 2 | 3 | | Q | 1 | 2 | | R | 4 | 5 | | S | 3 | 1 | | T | 5 | 4 | Calculate Spearman's Rank Correlation Coefficient between Judge A's and Judge B's rankings.
Solution 2:
1. Identify ranks: The data is already in ranks. Rank X = Judge A, Rank Y = Judge B.
2. Number of pairs (n): $n = 5$.
3. Calculate d, d2, and $\sum d^2$: | Participant | Rank X | Rank Y | d (Rank X - Rank Y) | d2 | | :---------- | :----- | :----- | :------------------ | :-- | | P | 2 | 3 | -1 | 1 | | Q | 1 | 2 | -1 | 1 | | R | 4 | 5 | -1 | 1 | | S | 3 | 1 | 2 | 4 | | T | 5 | 4 | 1 | 1 | | Total | | | 0 | 8 | $\sum d^2 = 8$.
4. Apply the formula: $$ \rho = 1 - \frac{6 \sum d^2}{n(n^2 - 1)} $$ $$ \rho = 1 - \frac{6 \times 8}{5(5^2 - 1)} $$ $$ \rho = 1 - \frac{48}{5(25 - 1)} $$ $$ \rho = 1 - \frac{48}{5(24)} $$ $$ \rho = 1 - \frac{48}{120} $$ $$ \rho = 1 - 0.4 $$ $$ \rho = 0.6 $$
Commentary: The Spearman's rank correlation coefficient is $0.6$. This indicates a moderately strong positive rank correlation between Judge A's and Judge B's rankings. There is a tendency for participants ranked highly in singing to also be ranked highly in dancing, though not perfectly. Question 3 (Calculating Spearman's Rank Correlation Coefficient from Raw Data - Without Ties): A small business owner recorded the number of years of experience (X) and monthly sales in thousands of Naira (Y) for 5 employees. | Employee | Years of Experience (X) | Monthly Sales (Y - ₦'000) | | :------- | :---------------------- | :-------------------------- | | 1 | 3 | 80 | | 2 | 7 | 150 | | 3 | 2 | 60 | | 4 | 10 | 180 | | 5 | 5 | 120 | Calculate Spearman's Rank Correlation Coefficient between years of experience and monthly sales.
Solution 3:
1. Rank X (Years of Experience): (Highest experience gets rank 1) 10 years (Rank 1) 7 years (Rank 2) 5 years (Rank 3) 3 years (Rank 4) 2 years (Rank 5)
2. Rank Y (Monthly Sales): (Highest sales gets rank 1) ₦180,000 Correlation is a statistical measure that quantifies the extent to which two variables are linearly related. It describes both the strength and the direction of the relationship between them.
Variables: In the context of correlation, variables are quantities that can take on different values. For example, "number of hours studied" and "exam score" are two variables. A scatter diagram (or scatter plot) is a graphical representation of the relationship between two variables. Each point on the diagram represents a pair of observations for the two variables.
Positive Correlation: Description: As the values of one variable increase, the values of the other variable also tend to increase. The points on the scatter diagram generally trend upwards from left to right.
Examples relevant to Nigeria: Amount of rainfall in a farming season and the yield of maize. Number of hours a student spends studying for JAMB and their score. Advertising expenditure by a company and its sales revenue.
Negative Correlation: Description: As the values of one variable increase, the values of the other variable tend to decrease. The points on the scatter diagram generally trend downwards from left to right.
Examples relevant to Nigeria: The price of petrol and the quantity consumed (assuming all else is equal). The number of years of education and the rate of unemployment. The speed of a vehicle and the time taken to cover a fixed distance.
Zero (or No)
Correlation: Description: There is no apparent linear relationship between the two variables. The points on the scatter diagram are scattered randomly, showing no clear upward or downward trend.
Examples relevant to Nigeria: A student's shoe size and their performance in a Further Mathematics exam. The amount of garri consumed by an individual and their height. The number of cars in Lagos and the average temperature in Kano.
Strength of Correlation: The closer the points on a scatter diagram cluster around a straight line, the stronger the correlation.
Strong Correlation: Points are tightly clustered along a line.
Weak Correlation: Points are widely scattered but still show a general trend.
Agriculture and Climate Science: Application: Farmers and agricultural scientists in Nigeria use correlation to understand the relationship between rainfall patterns, temperature, and crop yields (e.g., maize, rice, cassava). A positive correlation between rainfall and yield might suggest optimal planting seasons or the need for irrigation in dry periods. A negative correlation between temperature and a specific crop's yield could indicate crops sensitive to heat stress, informing variety selection.
Local Context: Studies could correlate soil nutrient levels (e.g., nitrogen, phosphorus) with the productivity of indigenous crops in different Nigerian states to guide fertiliser application and sustainable farming practices.
Economics and Business: Application: Economists and business analysts in Nigeria apply correlation to study relationships between economic indicators. For instance, they might correlate consumer spending with inflation rates, interest rates with investment levels, or crude oil prices with government revenue. Businesses use correlation to understand market trends, such as the relationship between advertising spend and sales revenue, or product price and quantity demanded.
Local Context: A Nigerian telecommunications company might correlate its promotional campaigns with new subscriber acquisition rates in different regions. A market researcher could correlate the price of staple foods (e.g., garri, beans) with the minimum wage to understand purchasing power and poverty levels.
Public Health and Social Development: Application: Public health officials use correlation to identify relationships between health interventions and outcomes. For example, they might correlate the number of vaccination campaigns in a state with the reduction in childhood diseases, or access to clean water facilities with a decrease in waterborne diseases like cholera. Social researchers can correlate literacy rates with economic development indicators across different communities.
Local Context: A non-governmental organization (NGO) working on maternal health in Nigeria could correlate the number of health awareness programs conducted in a rural community with the rate of facility-based deliveries. An urban planning department might correlate the number of available waste disposal facilities with the prevalence of environmental pollution-related illnesses in Lagos.