Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Calculate mean, median, mode, and range.
• Represent data using graphs.
• Determine the probability of simple events.
• Apply these concepts to solve problems.
• Analyze data and draw conclusions.

Lesson summary

Data handling and probability are essential skills in the modern world. Understanding how data is collected, organized, and interpreted allows us to make informed decisions based on evidence. In South Africa, data handling skills are crucial for understanding issues such as unemployment rates, crime statistics, health trends, and resource allocation. Probability helps us understand the likelihood of events and make predictions. For example, knowing the probability of rainfall can help farmers plan their planting schedules.

Lesson notes

Measures of Central Tendency Measures of central tendency describe the "average" or "typical" value in a data set.

There are three main measures: Mean: The sum of all values divided by the number of values. Also known as the average.

Formula: Mean = (Sum of all values) / (Number of values)

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

Mode: The value that appears most frequently in the data set. A data set can have no mode, one mode, or multiple modes.

Example 1: The ages of 7 learners in a Grade 9 class are: 14, 15, 14, 16, 15, 14, 15 Mean: (14 + 15 + 14 + 16 + 15 + 14 + 15) / 7 = 103/7 = 14.71 (approximately)

Median: First, arrange the data in ascending order: 14, 14, 14, 15, 15, 15,

1

6. The middle value is

1

5. Therefore, the median is

1

5. Mode: The value 14 appears 3 times, which is more than any other value.

Therefore, the mode is

1

4. Measures of Dispersion Measures of dispersion describe how spread out the data is.

We will focus on two measures: Range: The difference between the highest and lowest values in the data set.

Formula: Range = Highest Value - Lowest Value Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). The quartiles divide the data into four equal parts. Q1: The median of the lower half of the data. Q3: The median of the upper half of the data. IQR = Q3 - Q1 Example 2: Using the same data set of learner ages: 14, 15, 14, 16, 15, 14, 15 Range: The highest value is 16, and the lowest value is

1

4. Therefore, the range is 16 - 14 =

2. Interquartile Range (IQR): First, arrange the data in ascending order: 14, 14, 14, 15, 15, 15,

1

6. Q1: The median of the lower half (14, 14, 14) is

1

4. Q3: The median of the upper half (15, 15, 16) is

1

5. IQR = Q3 - Q1 = 15 - 14 =

1. Data Representation: Histograms and Box-and-Whisker Plots Histogram: A type of bar graph that shows the frequency distribution of continuous data. The bars touch each other, and the width of each bar represents a class interval.

Box-and-Whisker Plot: A graphical representation that displays the minimum value, first quartile (Q1), median (Q2), third quartile (Q3), and maximum value of a data set. It visually represents the spread and central tendency of the data. Probability Probability is the measure of how likely an event is to occur.

Theoretical Probability: The probability of an event based on mathematical reasoning and assumptions.

Formula: P(event) = (Number of favorable outcomes) / (Total number of possible outcomes)

Experimental Probability: The probability of an event based on the results of an experiment or observation.

Formula: P(event) = (Number of times the event occurred) / (Total number of trials)

Example 3: A bag contains 5 red balls and 3 blue balls. What is the probability of randomly selecting a red ball? (Theoretical Probability) Number of favorable outcomes (red balls) = 5 Total number of possible outcomes (total balls) = 5 + 3 = 8 P(red ball) = 5/8 Example 4: A coin is tossed 100 times. Heads appears 55 times. What is the experimental probability of getting heads? Number of times the event occurred (heads) = 55 Total number of trials (tosses) = 100 P(heads) = 55/100 = 0.55 Guided Practice (With Solutions)

Question 1: The marks (out of 50) of 10 learners in a Mathematics test are: 35, 40, 42, 38, 45, 32, 40, 48, 35,

3

5. Calculate the mean, median, and mode of the marks.

Solution: Mean: (35 + 40 + 42 + 38 + 45 + 32 + 40 + 48 + 35 + 35) / 10 = 380 / 10 = 38 Median: First, arrange the data in ascending order: 32, 35, 35, 35, 38, 40, 40, 42, 45,

4

8. The two middle values are 38 and

4

0. Therefore, the median is (38 + 40) / 2 =

3

9. Mode: The value 35 appears 3 times, which is more than any other value.

Therefore, the mode is

3

5. Commentary: Remember to arrange the data in ascending order before finding the median. The mode is simply the most frequent value.

Question 2: Using the same data from Question 1, calculate the range and interquartile range.

Solution: Range: The highest mark is 48, and the lowest mark is

3

2. Therefore, the range is 48 - 32 =

1

6. Interquartile Range (IQR): The data in ascending order is: 32, 35, 35, 35, 38, 40, 40, 42, 45,

4

8. Q1: The median of the lower half (32, 35, 35, 35, 38) is

3

5. Q3: The median of the upper half (40, 40, 42, 45, 48) is

4

2. IQR = Q3 - Q1 = 42 - 35 =

7. Commentary: The range gives a simple measure of the overall spread, while the IQR focuses on the spread of the middle 50% of the data, making it less sensitive to outliers.

Question 3: A spinner has 6 equally likely sections numbered 1 to

6. What is the probability of spinning a number greater than 4?

Solution: Number of favorable outcomes (5 or 6) = 2 Total number of possible outcomes (1, 2, 3, 4, 5, or 6) = 6 P(number greater than 4) = 2/6 = 1/3

Commentary: Make sure to identify all the outcomes that satisfy the condition (greater than 4).