Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate mean, median, and mode.
• Determine range, quartiles, and interquartile range (IQR).
• Interpret these measures in real-world contexts.
• Identify outliers and understand their impact.
• Choose the best measure (mean, median, or mode) based on data distribution.

Lesson summary

Data handling is a crucial skill in the 21st century, especially in a diverse and developing country like South Africa. We are constantly bombarded with information, from news reports on unemployment rates to statistics on crime, electricity usage, and health trends. Understanding how to summarise and interpret this data allows us to make informed decisions about our lives, participate meaningfully in our communities, and critically evaluate information presented to us. For example, understanding crime statistics in your area can influence decisions about security measures for your home. Analysing unemployment rates can help you understand job market trends and inform your career choices.

Lesson notes

2.1 Measures of Central Tendency Measures of central tendency describe the 'center' of a dataset.

We will focus on three key measures: Mean: The average of all the values in a dataset. To calculate the mean, we sum all the values and divide by the number of values.

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

Example: Imagine we want to find the average household size in a small rural village. We survey 10 households and find the following number of people per household: 3, 4, 2, 5, 6, 3, 4, 4, 2,

5. Mean = (3 + 4 + 2 + 5 + 6 + 3 + 4 + 4 + 2 + 5) / 10 = 38 / 10 = 3.8 So, the average household size in this village is 3.8 people.

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

Example: Using the same household data, first we arrange the numbers in ascending order: 2, 2, 3, 3, 4, 4, 4, 5, 5,

6. Since there are 10 (even) numbers, the median is the average of the 5th and 6th numbers: (4 + 4) / 2 = 4 So, the median household size is 4 people.

Mode: The value that appears most frequently in a dataset. A dataset can have no mode (if all values appear only once), one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).

Example: In our household data (2, 2, 3, 3, 4, 4, 4, 5, 5, 6), the number 4 appears three times, which is more than any other number.

Therefore, the mode is 4. 2.2 Measures of Spread Measures of spread describe how 'spread out' the data is. We will focus on the range and interquartile range (IQR).

Range: The difference between the largest and smallest values in a dataset.

Formula: Range = Maximum value - Minimum value

Example: For our household data (2, 2, 3, 3, 4, 4, 4, 5, 5, 6), the maximum value is 6 and the minimum value is

2. Range = 6 - 2 = 4 The range is 4, indicating the spread of household sizes.

Quartiles: Quartiles divide a dataset into four equal parts when the data is arranged in ascending order.

Q1 (First Quartile): The median of the lower half of the data. 25% of the data falls below Q

1. Q2 (Second Quartile): The median of the entire dataset. This is the same as the median we calculated earlier. 50% of the data falls below Q

2. Q3 (Third Quartile): The median of the upper half of the data. 75% of the data falls below Q

3. Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1). It represents the range of the middle 50% of the data.

Formula: IQR = Q3 - Q1

Example: Using the household data (2, 2, 3, 3, 4, 4, 4, 5, 5, 6): The median (Q2) is 4 (as we calculated before).

The lower half of the data is: 2, 2, 3, 3,

4. The median of this lower half (Q1) is

3. The upper half of the data is: 4, 4, 5, 5,

6. The median of this upper half (Q3) is

5. IQR = Q3 - Q1 = 5 - 3 = 2 The IQR is 2, meaning the middle 50% of the household sizes vary within a range of 2. 2.3 Outliers An outlier is a data point that is significantly different from other data points in a dataset. Outliers can skew the mean and affect the range.

Identifying Outliers: A common method for identifying outliers uses the IQR: Lower Bound: Q1 - 1.5 IQR Upper Bound: Q3 + 1.5 IQR Any data point below the Lower Bound or above the Upper Bound is considered an outlier.

Example: Using our household data, let's add an outlier: 2, 2, 3, 3, 4, 4, 4, 5, 5, 6,

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5. The "15" is a much larger household and might be a multi-generational household. Q1 = 3 Q3 = 5 IQR = 2 Lower Bound = 3 - 1.5 2 = 0 Upper Bound = 5 + 1.5 2 = 8 Since 15 is greater than 8, it is an outlier. The outlier significantly increases the mean and range. 2.4 Comparing Mean, Median and Mode Symmetric Distribution: When the data is evenly distributed around the center, the mean, median, and mode are approximately equal.

Skewed Distribution: Right Skew (Positive Skew): The tail of the distribution is longer on the right side. The mean is greater than the median, which is greater than the mode. This typically happens when there are few very high values.

Left Skew (Negative Skew): The tail of the distribution is longer on the left side. The mean is less than the median, which is less than the mode. This typically happens when there are few very low values. In a skewed distribution, the median is often a better measure of central tendency than the mean because it is less affected by outliers. The mode is less reliable as a measure of central tendency because it only reflects the most frequent value. Guided Practice (With Solutions)

Question 1: A survey of weekly wages (in Rands) of 7 domestic workers in a neighborhood yielded the following data: 800, 900, 850, 1000, 900, 800,

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0. Calculate the mean weekly wage.

Solution: Mean = (800 + 900 + 850 + 1000 + 900 + 800 + 850) / 7 = 6100 / 7 = 871.43 (rounded to two decimal places) The mean weekly wage is R871.

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3. Question 2: The ages of 9 participants in an adult literacy program are: 22, 25, 30, 45, 28, 32, 50, 25, 35.