Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Calculate and interpret mean, median, mode, and range.
• Represent data using histograms, box plots, and scatter plots.
• Calculate probabilities of single and combined events.
• Apply problem-solving to exam-style questions.
• Understand biased vs. unbiased sampling.

Lesson summary

This week, we delve deeper into data handling and probability, consolidating the concepts we introduced last week and focusing on skills necessary for exam preparation. Data handling and probability are crucial skills for understanding the world around us. From interpreting crime statistics in news reports to making informed decisions about insurance policies or predicting election outcomes, these mathematical tools are essential for navigating everyday life as informed citizens in South Africa. Understanding probability can help us assess risks, make better financial decisions, and even understand the randomness inherent in games of chance like the National Lottery.

Lesson notes

2.1 Measures of Central Tendency These measures describe the "center" of a dataset.

Mean: The average of all the values. Calculated by summing all the values and dividing by the number of values. _Mean = (Sum of all values) / (Number of values)_ Example (Ungrouped): The ages of five learners are 14, 15, 14, 16,

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5. The mean age is (14 + 15 + 14 + 16 + 15) / 5 = 74 / 5 = 14.8 years.

Example (Grouped): Consider a frequency table showing the number of hours Grade 9 learners spend on homework per week: | Hours | Frequency | | -------- | --------- | | 0-2 | 5 | | 2-4 | 10 | | 4-6 | 8 | | 6-8 | 2 | To calculate the mean, we first find the midpoint of each interval (e.g., (0+2)/2 = 1 for the first interval). Then we multiply the midpoint by the frequency for each interval, sum these products, and divide by the total frequency.

Midpoints: 1, 3, 5, 7 Calculations: (1 5) + (3 10) + (5 8) + (7 2) = 5 + 30 + 40 + 14 = 89 Total Frequency: 5 + 10 + 8 + 2 = 25 Mean: 89 / 25 = 3.56 hours.

Median: The middle value when the data is arranged in ascending order. If there are two middle values, the median is the average of these two.

Example (Ungrouped): The scores on a test are: 60, 70, 50, 80,

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0. First, order the data: 50, 60, 70, 80,

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0. The median is

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0. Example (Ungrouped with even number of values): The scores on a test are: 60, 70, 50, 80, 90,

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0. First, order the data: 40, 50, 60, 70, 80,

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0. The median is (60+70)/2 =

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5. Mode: The value that appears most frequently. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).

Example (Ungrouped): The colours of cars in a parking lot are: Red, Blue, Red, Green, Red, Blue, Black. The mode is Red. 2.2 Measures of Dispersion These measures describe how spread out the data is.

Range: The difference between the highest and lowest values. _Range = Highest value - Lowest value_

Example: The ages of learners are 14, 15, 14, 16,

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5. The range is 16 - 14 = 2 years.

Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). The quartiles divide the ordered data into four equal parts. Q1 is the median of the lower half of the data, and Q3 is the median of the upper half. _IQR = Q3 - Q1_

Example: The following data represents the heights (in cm) of 11 Grade 9 learners: 145, 150, 152, 155, 158, 160, 162, 165, 168, 170,

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2. Q2 (Median) = 160 Q1 (Median of lower half): 145, 150, 152, 155,

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8. Q1 = 152 Q3 (Median of upper half): 162, 165, 168, 170,

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2. Q3 = 168 IQR = 168 - 152 = 16 cm 2.3 Data Representation Histograms: Used to represent grouped data. The bars touch each other, and the area of each bar is proportional to the frequency of the corresponding class interval.

Box-and-Whisker Plots (Boxplots): Visually represent the median, quartiles, and extreme values (minimum and maximum) of a dataset. They are useful for comparing distributions and identifying outliers.

Scatter Plots: Used to display the relationship between two numerical variables. Each point on the plot represents a pair of values. 2.4 Probability Probability of a single event: The ratio of the number of favorable outcomes to the total number of possible outcomes. _P(event) = (Number of favorable outcomes) / (Total number of possible outcomes)_

Example: What is the probability of rolling a 4 on a fair six-sided die? There is one favorable outcome (rolling a 4) and six possible outcomes (1, 2, 3, 4, 5, 6).

Therefore, P(rolling a 4) = 1/

6. Combined Events: Independent Events: The outcome of one event does not affect the outcome of the other. The probability of both events occurring is the product of their individual probabilities. _P(A and B) = P(A) P(B)_

Example: Flipping a coin twice. The outcome of the first flip does not affect the outcome of the second flip. The probability of getting heads on both flips is (1/2) (1/2) = 1/

4. Dependent Events: The outcome of one event does affect the outcome of the other. The probability of the second event depends on the outcome of the first.

Example: Drawing two cards from a deck without replacement. The probability of drawing a second card of a certain type is affected by what was drawn first.

Tree Diagrams: Useful for visualizing and calculating probabilities of combined events, especially when the events are sequential.

Two-Way Tables: Useful for organizing and calculating probabilities based on categorical data. 2.5 Sampling Techniques and Bias Biased Sample: A sample that does not accurately represent the population. This can lead to incorrect conclusions.

Example: Surveying only wealthy residents about their opinions on a proposed tax increase will likely produce a biased result, as their interests may differ from those of the general population.

Unbiased Sample: A sample that accurately represents the population. Random sampling is a common method for obtaining an unbiased sample.

Example: Drawing names from a hat to select participants for a survey.