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).
• Calculate and interpret measures of spread (range, interquartile range).
• Construct and interpret box and whisker plots.
• Analyse data from tables, charts, and graphs.
• Use data to make informed decisions and predictions.

Lesson summary

Data handling is a crucial skill in the 21st century, and particularly important in South Africa. We are constantly bombarded with information – from election results to crime statistics, from weather forecasts to financial news. Being able to summarise this data effectively and interpret it accurately is essential for making informed decisions in our daily lives and participating meaningfully in our society. Whether it's understanding the impact of load shedding on local businesses or analysing the spread of diseases like COVID-19, data skills are vital for navigating the complexities of modern South African life.

Lesson notes

2.1 Measures of Central Tendency: Mean: The average of a set of numbers. It is calculated by adding up all the values and dividing by the total number of values.

Ungrouped data:* Mean = (Sum of all values) / (Number of values)

Grouped data: Mean ≈ (Sum of (Midpoint of interval Frequency)) / (Total Frequency) We use midpoints of class intervals as approximations for the data within each interval. This is because we do not know the exact values within the intervals themselves, only the number of values which lie in the interval.

Median: The middle value in a set of data when the data is arranged in ascending order.

Ungrouped data:* If there are an odd number of values, the median is the middle value. If there are an even number of values, the median is the average of the two middle values.

Grouped data: We identify the median class (the class interval containing the middle value). The median can then be estimated using a formula (not generally required at this level) or by making an approximation using the cumulative frequency. We identify which class interval has the median value by looking at the cumulative frequency.

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

Ungrouped data:* Simply identify the value that appears most often.

Grouped data:* The modal class is the class interval with the highest frequency. The mode is approximated by the midpoint of the modal class.

Example 1: Calculating Mean, Median, and Mode (Ungrouped Data) The ages of 7 students in a Grade 11 class are: 16, 17, 16, 18, 17, 17, 15 Mean: (16 + 17 + 16 + 18 + 17 + 17 + 15) / 7 = 116 / 7 = 16.57 years (rounded to two decimal places)

Median: First, arrange the data in ascending order: 15, 16, 16, 17, 17, 17,

1

8. The middle value is

1

7. So, the median age is 17 years.

Mode: The age 17 appears most frequently (3 times). So, the mode is 17 years.

Example 2: Calculating Mean, Median Class, and Modal Class (Grouped Data) The table shows the number of hours 30 students spend on social media per week: | Hours | Frequency | | :----- | :-------- | | 0 - 5 | 5 | | 5 - 10 | 8 | | 10 - 15| 10 | | 15 - 20| 7 | Approximate Mean: Midpoints: 2.5, 7.5, 12.5, 17.5 (2.5 5) + (7.5 8) + (12.5 10) + (17.5 * 7) = 12.5 + 60 + 125 + 122.5 = 320 Mean ≈ 320 / 30 = 10.67 hours (rounded to two decimal places)

Modal Class: The class interval with the highest frequency is 10 - 15 hours.

Median Class: The median is the value in the middle. Since there are 30 data points, the median is the average of the 15th and 16th value. Calculate Cumulative Frequency. | Hours | Frequency | Cumulative Frequency | | :----- | :-------- | :-------------------- | | 0 - 5 | 5 | 5 | | 5 - 10 | 8 | 13 | | 10 - 15| 10 | 23 | | 15 - 20| 7 | 30 | The 13th value is in the 5 - 10 interval. The 23rd value is in the 10 - 15 interval.

Therefore, the 15th and 16th values are in the 10-15 interval, and that is the median class. 2.2 Measures of Spread: Range: The difference between the highest and lowest values in a dataset. Range = Maximum Value - Minimum Value.

Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). It represents the spread of the middle 50% of the data. IQR = Q3 - Q

1. Quartiles:* Quartiles divide the data into four equal parts. Q1 is the first quartile (25th percentile), Q2 is the second quartile (50th percentile, which is also the median), and Q3 is the third quartile (75th percentile).

Example 3: Calculating Range and Interquartile Range Using the same ages from Example 1: 15, 16, 16, 17, 17, 17, 18 Range: 18 - 15 = 3 years Interquartile Range: Q1 (Lower Quartile): The median of the lower half of the data (excluding the overall median if the number of data points is odd).

Lower half: 15, 16,

1

6. Q1 = 16 Q3 (Upper Quartile): The median of the upper half of the data (excluding the overall median if the number of data points is odd).

Upper half: 17, 17,

1

8. Q3 = 17 IQR = Q3 - Q1 = 17 - 16 = 1 year 2.3 Box and Whisker Plots: A box and whisker plot is a graphical representation of data that shows the minimum value, first quartile (Q1), median (Q2), third quartile (Q3), and maximum value. It visually represents the spread and central tendency of the data. Box-and-whisker plots are useful for visually comparing two or more sets of data.

Construction: Draw a number line that covers the range of the data. Mark the minimum value, Q1, median, Q3, and maximum value on the number line. Draw a box from Q1 to Q

3. Draw a vertical line inside the box at the median. Draw whiskers from the box to the minimum and maximum values.

Outliers: Values that fall far outside the rest of the data. Outliers can be identified using the following rule: Lower Outlier: Value Q3 + 1.5 IQR If outliers exist, the whiskers extend to the most extreme non-outlier data point. Outliers are marked separately as points.