Lesson Notes By Weeks and Term v5 - Grade 10
Statistics – Week 4 focus
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Statistics – Week 4 focus
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Subject: Mathematics
Class: Grade 10
Term: 3rd Term
Week: 4
Theme: General lesson support
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Statistics is a crucial branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. In Grade 10, understanding statistics empowers you to make informed decisions based on evidence. This week, we will be focusing on measures of central tendency and dispersion, specifically how to calculate and interpret the mean, median, mode, range, interquartile range (IQR), and standard deviation for both ungrouped and grouped data. This knowledge is essential for interpreting social trends, economic indicators, and scientific findings relevant to South Africa.
2.1 Measures of Central Tendency: Measures of central tendency describe the "typical" value in a dataset.
Mean (Average): The sum of all the values divided by the number of values.
Ungrouped Data:* Mean (x̄) = (Σx) / n, where Σx is the sum of all values and n is the number of values.
Grouped Data: Mean (x̄) ≈ (Σ(f m)) / Σf, where f is the frequency of each interval and m is the midpoint of each interval.
Example (Ungrouped): The ages of five learners are 15, 16, 15, 17, and
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6. The mean age is (15 + 16 + 15 + 17 + 16) / 5 = 79 / 5 = 15.8 years.
Example (Grouped): Consider the following table showing the ages of 40 learners: | Age Group | Frequency (f) | Midpoint (m) | f * m | | --------- | ------------- | ------------- | ----- | | 14-15 | 10 | 14.5 | 145 | | 16-17 | 20 | 16.5 | 330 | | 18-19 | 10 | 18.5 | 185 | | Total | 40 | | 660 | The estimated mean age is 660 / 40 = 16.5 years. Notice we are estimating the mean because we do not have the individual ages, only the groups.
Median: The middle value when the data is arranged in ascending order.
Ungrouped Data:* If n is odd, the median is the (n+1)/2 th value. If n is even, the median is the average of the n/2 th and (n/2 + 1) th values.
Grouped Data: The median lies within the median class. To find the median, we use linear interpolation: Median = L + [(n/2 - cf) / f m ] w , where L is the lower boundary of the median class, n is the total frequency, cf is the cumulative frequency of the class before the median class, f m is the frequency of the median class, and w is the class width.
Example (Ungrouped): The marks of seven students are: 60, 70, 50, 80, 90, 75,
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5. Arranging in ascending order: 50, 60, 65, 70, 75, 80,
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0. The median is
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0. Example (Grouped): Consider the table below with cumulative frequency: | Age Group | Frequency (f) | Cumulative Frequency (cf) | | --------- | ------------- | ------------------------- | | 14-15 | 10 | 10 | | 16-17 | 20 | 30 | | 18-19 | 10 | 40 | | Total | 40 | | The median lies in the age group 16-17 since n/2 = 40/2 = 20, and the cumulative frequency just exceeds 20 in the 16-17 group.
Therefore, L = 15.5 (lower boundary), n = 40, cf = 10, f m = 20, and w =
2. Median = 15.5 + [(20-10) / 20] * 2 = 15.5 + 1 = 16.5 Mode: The value that appears most frequently in a dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).
Ungrouped Data:* Simply identify the most frequent value.
Grouped Data: The modal class is the class with the highest frequency. The mode can be approximated by the midpoint of the modal class. A more accurate estimate can be obtained using the formula: Mode = L + [ (f m - f m-1 ) / (2f m - f m-1 - f m+1 ) ] w , where L is the lower boundary of the modal class, f m is the frequency of the modal class, f m-1 is the frequency of the class before the modal class, f m+1 is the frequency of the class after the modal class, and w is the class width.
Example (Ungrouped): The number of siblings for eight learners are: 1, 2, 0, 1, 3, 1, 2,
4. The mode is 1 (appears three times).
Example (Grouped): In the age group table above, the modal class is 16-17 because it has the highest frequency (20).
Therefore, L = 15.5, f m = 20, f m-1 = 10, f m+1 = 10, and w =
2. Mode = 15.5 + [(20 - 10) / (220 - 10 - 10)] 2 = 15.5 + (10 / 20) * 2 = 15.5 + 1 = 16.5 2.2 Measures of Dispersion: Measures of dispersion describe the spread or variability of data.
Range: The difference between the highest and lowest values in a dataset. Range = Maximum value - Minimum value.
Example: For the ages 15, 16, 15, 17, 16, the range is 17 - 15 = 2 years.
Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). IQR = Q3 - Q
1. The IQR represents the spread of the middle 50% of the data.
Ungrouped Data:* Q1 is the median of the lower half of the data, and Q3 is the median of the upper half of the data.
Grouped Data:* Similar to the median, we use linear interpolation to find Q1 and Q
3. Q1 lies in the class where the cumulative frequency first exceeds n/4, and Q3 lies in the class where the cumulative frequency first exceeds 3n/
4. The formulas are: Q1 = L Q1 + [(n/4 - cf Q1 ) / f Q1 ] w Q3 = L Q3 + [(3n/4 - cf Q3 ) / f Q3 ] w Where: L Q1 and L Q3 are the lower boundaries of the Q1 and Q3 classes respectively, cf Q1 and cf Q3 are the cumulative frequencies before the Q1 and Q3 classes, and f Q1 and f Q3 are the frequencies of the Q1 and Q3 classes.
Example (Ungrouped): The following are test scores of 11 learners: 40, 50, 55, 60, 65, 70, 75, 80, 85, 90,
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5. Q1: (n+1)/4 = (11+1)/4 =
3. The third value is
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5. So Q1 =
5
5. Q3: 3(n+1)/4 = 3(11+1)/4 =
9. The ninth value is
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5. So Q3 =
8
5. IQR = Q3 - Q1 = 85 - 55 = 30.