Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate and interpret measures of central tendency (mean, median, mode).
• Determine and interpret measures of spread (range, interquartile range).
• Interpret different data displays like bar graphs and histograms.
• Critically evaluate data presented in various contexts.

Lesson summary

Data handling is a crucial skill in Mathematical Literacy. In Week 4, we focus on summarising and interpreting data, building upon the foundational concepts covered earlier. Understanding data presentation and being able to draw meaningful conclusions is incredibly important in everyday life. From understanding the cost of living increases reported in the news to making informed choices about cellular data packages or even understanding the spread of disease outbreaks, the ability to interpret data empowers you to make sound decisions.

Lesson notes

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

The three main measures are: Mean: The sum of all the values divided by the number of values. This is what most people think of when they hear the word "average".

Formula (Ungrouped Data): Mean (x̄) = Σx / n, where Σx is the sum of all values and n is the number of values.

Formula (Grouped Data): Mean (x̄) = Σ(f m) / Σf, where f is the frequency of each class interval and m is the midpoint of the class interval. Remember to find the midpoint by adding the upper and lower boundaries of the interval and dividing by

2. 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 dataset. A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).

Example 1: Calculating Mean, Median, and Mode (Ungrouped Data) A small spaza shop in Khayelitsha records the number of airtime vouchers sold each day for a week: 15, 18, 20, 15, 22, 25,

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5. Mean: (15 + 18 + 20 + 15 + 22 + 25 + 15) / 7 = 130 / 7 = 18.57 (approximately 19 vouchers)

Median: First, order the data: 15, 15, 15, 18, 20, 22,

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5. The middle value is

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8. Mode: The value 15 appears most frequently (3 times).

Example 2: Calculating Mean (Grouped Data) The following table shows the distribution of monthly salaries of workers in a clothing factory in Durban: | Salary Range (ZAR) | Frequency | |----------------------|-----------| | 3000 - 4000 | 10 | | 4000 - 5000 | 25 | | 5000 - 6000 | 35 | | 6000 - 7000 | 20 | | 7000 - 8000 | 10 | Find the midpoint (m) of each class interval: 3000 - 4000: (3000 + 4000) / 2 = 3500 4000 - 5000: (4000 + 5000) / 2 = 4500 5000 - 6000: (5000 + 6000) / 2 = 5500 6000 - 7000: (6000 + 7000) / 2 = 6500 7000 - 8000: (7000 + 8000) / 2 = 7500 Multiply the frequency (f) by the midpoint (m) for each class: 10 3500 = 35000 25 4500 = 112500 35 5500 = 192500 20 6500 = 130000 10 7500 = 75000 Sum the results (Σ(f * m)) and the frequencies (Σf): Σ(f m) = 35000 + 112500 + 192500 + 130000 + 75000 = 545000 Σf = 10 + 25 + 35 + 20 + 10 = 100 Calculate the mean: Mean = 545000 / 100 = 5450 ZAR Measures of Spread Measures of spread describe how dispersed or varied the data is.

Range: The difference between the highest and lowest values in the dataset. Simple but sensitive to outliers.

Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). Q1 is the median of the lower half of the data, and Q3 is the median of the upper half of the data. The IQR represents the spread of the middle 50% of the data, making it less sensitive to outliers.

Example 3: Calculating Range and IQR Using the airtime voucher data from Example 1: 15, 18, 20, 15, 22, 25,

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5. Range: 25 - 15 = 10 IQR: Order the data: 15, 15, 15, 18, 20, 22, 25 Find the median: 18 Q1 (median of the lower half: 15, 15, 15): 15 Q3 (median of the upper half: 20, 22, 25): 22 IQR = Q3 - Q1 = 22 - 15 = 7 Data Displays Different types of data displays are used to visually represent data: Histograms: Used for continuous data. Bars touch each other, representing the frequency of data within intervals.

Bar Graphs: Used for categorical data. Bars are separated, representing the frequency or amount of each category.

Pie Charts: Used to show proportions of a whole. Each slice represents a percentage of the total.

Line Graphs: Used to show trends over time. Data points are connected by lines.

Box-and-Whisker Plots: Shows the median, quartiles (Q1 and Q3), and minimum/maximum values. Excellent for comparing the distribution of different datasets. The box represents the IQR, and the whiskers extend to the minimum and maximum values within a certain range (usually 1.5 times the IQR).

Interpreting Data Displays: Look for patterns, trends, and outliers. Pay attention to the labels, scales, and units. Consider the source of the data and potential biases.

Example 4: Interpreting a Bar Graph A bar graph shows the number of reported crimes in different suburbs of Cape Town in

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3. Suburb A has the highest bar, followed by Suburb B, and then Suburb C. This indicates that Suburb A had the highest number of reported crimes in

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3. Further analysis might look at the types of crimes in each suburb. Guided Practice (With Solutions)

Question 1: The following data represents the ages of 10 learners in a Grade 11 Mathematical Literacy class: 16, 17, 16, 18, 17, 16, 17, 18, 19,

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7. Calculate the mean, median, and mode.

Solution: Mean: (16 + 17 + 16 + 18 + 17 + 16 + 17 + 18 + 19 + 17) / 10 = 171 / 10 = 17.1 years Median: First, order the data: 16, 16, 16, 17, 17, 17, 17, 18, 18,

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9. Since there are 10 values, the median is the average of the 5th and 6th values: (17 + 17) / 2 = 17 years Mode: The value 17 appears most frequently (4 times).