Lesson Notes By Weeks and Term v5 - Grade 10
Statistics – Week 5 focus
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Statistics – Week 5 focus
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Subject: Mathematics
Class: Grade 10
Term: 3rd Term
Week: 5
Theme: General lesson support
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This week, we delve deeper into the world of statistics, focusing on measures of dispersion: range, interquartile range (IQR), variance, and standard deviation. Understanding these concepts is crucial for interpreting data accurately and making informed decisions. In South Africa, statistics is used extensively in various sectors, from monitoring crime rates to analyzing economic trends and evaluating the effectiveness of social programs. For instance, understanding the spread of income inequality requires analyzing the range and standard deviation of income data. Similarly, predicting the spread of a disease like HIV/AIDS depends on statistical models that consider data variability.
2.1 Range: The range is the simplest measure of dispersion. It's 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 10 class: 15, 16, 15, 17, 16, 15, 18,
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6. Maximum Value = 18 Minimum Value = 15 Range = 18 - 15 = 3 The range of ages is 3 years. While simple, it is heavily affected by outliers. 2.2 Interquartile Range (IQR): The IQR measures the spread of the middle 50% of the data. It is more resistant to outliers than the range. To calculate the IQR, we first need to find the quartiles.
Q1 (First Quartile or 25th percentile): The value below which 25% of the data lies.
Q2 (Second Quartile or 50th percentile): The median of the data.
Q3 (Third Quartile or 75th percentile): The value below which 75% of the data lies.
Formula: IQR = Q3 - Q1
Example: Let's use the following test scores from a class: 50, 60, 65, 70, 75, 80, 85, 90, 95,
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0. Order the data: 50, 60, 65, 70, 75, 80, 85, 90, 95, 100 Find the median (Q2): (75+80)/2 = 77.5 Find Q1: The median of the lower half (50, 60, 65, 70, 75) is
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5. Find Q3: The median of the upper half (80, 85, 90, 95, 100) is
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0. Calculate IQR: IQR = 90 - 65 = 25 The interquartile range is
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5. This means the middle 50% of the test scores are spread over a range of 25 marks. 2.3 Variance: Variance measures the average squared difference between each data point and the mean. A higher variance indicates greater spread.
Formula (Ungrouped Data): First, calculate the mean: μ = (∑x i ) / n, where x i are individual data points and n is the number of data points. Then, calculate the variance: σ 2 = ∑(x i - μ) 2 / n Formula (Grouped Data): First, calculate the mean: μ = (∑f i x i ) / ∑f i , where x i is the midpoint of each interval and f i is the frequency of that interval. Then, calculate the variance: σ 2 = ∑f i (x i - μ) 2 / ∑f i (
Note: Sometimes you will see (n-1) used in the denominator of the ungrouped variance formula. This is the "sample variance" and gives an unbiased estimate of the population variance. For Grade 10 CAPS, using 'n' is typically sufficient.)
Example (Ungrouped Data): Consider the number of siblings for 5 students: 1, 2, 0, 3,
1. Calculate the mean: μ = (1 + 2 + 0 + 3 + 1) / 5 = 7/5 = 1.4 Calculate the squared differences: (1 - 1.4) 2 = 0.16 (2 - 1.4) 2 = 0.36 (0 - 1.4) 2 = 1.96 (3 - 1.4) 2 = 2.56 (1 - 1.4) 2 = 0.16 Sum the squared differences: 0.16 + 0.36 + 1.96 + 2.56 + 0.16 = 5.2 Divide by the number of data points: σ 2 = 5.2 / 5 = 1.04 The variance is 1.
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4. Example (Grouped Data): The following table shows the marks obtained by 40 students in a test: | Marks | Frequency (f i ) | Midpoint (x i ) | f i x i | (x i - μ) | (x i - μ) 2 | f i (x i - μ) 2 | |-----------|-----------------------|-----------------------|-------------------|--------------------|-----------------------|---------------------------------| | 40-50 | 4 | 45 | 180 | -22.5 | 506.25 | 2025 | | 50-60 | 8 | 55 | 440 | -12.5 | 156.25 | 1250 | | 60-70 | 12 | 65 | 780 | -2.5 | 6.25 | 75 | | 70-80 | 10 | 75 | 750 | 7.5 | 56.25 | 562.5 | | 80-90 | 6 | 85 | 510 | 17.5 | 306.25 | 1837.5 | | Total | 40 | | 2660 | | | 5750 | Calculate the mean: μ = 2660 / 40 = 66.5 Calculate the variance: σ 2 = 5750 / 40 = 143.75 The variance is 143.75. 2.4 Standard Deviation: The standard deviation is the square root of the variance. It provides a more interpretable measure of spread because it is in the same units as the original data.
Formula: Standard Deviation (σ) = √Variance (σ 2 ) Example (Using previous Ungrouped Data Example): We found the variance for the number of siblings to be 1.
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4. Standard Deviation = √1.04 ≈ 1.02 The standard deviation is approximately 1.02 siblings. This tells us that, on average, the number of siblings each student has deviates by about 1 sibling from the mean. Example (Using previous Grouped Data Example): We found the variance for the test marks to be 143.
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5. Standard Deviation = √143.75 ≈ 11.99 The standard deviation is approximately 11.
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9. This tells us that, on average, a student's mark deviates by about 12 marks from the mean. Guided Practice (With Solutions)
Question 1: The heights (in cm) of 7 learners are: 150, 155, 160, 162, 165, 170,
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5. Calculate the range.
Solution: Maximum Height = 175 cm Minimum Height = 150 cm Range = 175 - 150 = 25 cm
Commentary: This question directly applies the range formula. Understanding the concept of 'maximum' and 'minimum' is essential.* Question 2: The following data represents the number of hours spent on homework per week by 9 learners: 5, 7, 8, 10, 12, 13, 15, 16,
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8. Calculate the interquartile range (IQR).
Solution: Data is already ordered. Median (Q2) = 12 Lower half: 5, 7, 8, 10 Q1 = (7+8)/2 = 7.5 Upper half: 13, 15, 16, 18 Q3 = (15+16)/2 = 15.5 IQR = Q3 - Q1 = 15.5 - 7.5 = 8
Commentary: This question requires identifying the quartiles.