Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Calculate mean, median, and mode for ungrouped and grouped data.
• Determine measures of dispersion like range and IQR.
• Interpret the meaning of these measures.
• Choose the appropriate measure for a given dataset.

Lesson summary

Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data. It provides us with the tools to make informed decisions and draw meaningful conclusions from information. In a South African context, understanding statistics is crucial for interpreting news reports about unemployment rates, analyzing crime statistics, evaluating the effectiveness of public health campaigns (like those related to HIV/AIDS or COVID-19), and understanding economic trends affecting our communities.

Furthermore, many fields, from market research to agriculture, require statistical literacy.

Lesson notes

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

The three main measures are: Mean: The sum of all values divided by the number of values. It's the most common average.

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

Grouped Data: Mean (x̄) = Σ(f x) / Σf, where f is the frequency of each class interval and x is the midpoint of each class interval.

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. It is less affected by outliers than the mean.

Ungrouped Data: Arrange the data in ascending order. If n is odd, the median is the (n+1)/2 th value. If n is even, the median is the average of the n/2 th and (n/2 + 1) th values.

Grouped Data: Median = L + [(n/2 - CF)/f] w, where L is the lower boundary of the median class, n is the total frequency, CF is the cumulative frequency of the class before the median class, f is the frequency of the median class, and w is the class width. First, identify the median class (the class containing the n/2 th value).

Mode: The value that appears most frequently in the dataset. There can be more than one mode (bimodal, multimodal), or no mode at all. The mode is useful for categorical data but less so for continuous data.

Ungrouped Data: Simply identify the value that appears most often.

Grouped Data: Modal Class: The class with the highest frequency. The Mode can be approximated as the midpoint of the Modal Class. A more precise estimate can be made using the formula: Mode = L + [(fm - fm-1) / (2fm - fm-1 - fm+1)] w, where L is the lower boundary of the modal class, fm is the frequency of the modal class, fm-1 is the frequency of the class before the modal class, fm+1 is the frequency of the class after the modal class and w is the class width.

Example 1 (Mean - Ungrouped Data): The ages of 5 students in a Grade 11 class are: 16, 17, 16, 18,

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7. Calculate the mean age.

Solution: Mean = (16 + 17 + 16 + 18 + 17) / 5 = 84 / 5 = 16.8 years Example 2 (Median - Ungrouped Data): The number of hours spent on homework by 7 students in a week are: 5, 3, 8, 2, 6, 3,

7. Find the median number of hours.

Solution: First, arrange the data in ascending order: 2, 3, 3, 5, 6, 7,

8. Since there are 7 values (odd number), the median is the (7+1)/2 = 4th value, which is

5. Therefore, the median is 5 hours.

Example 3 (Mode - Ungrouped Data): The shoe sizes of 10 learners are: 6, 7, 8, 7, 6, 9, 7, 8, 7,

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0. What is the mode?

Solution: The shoe size 7 appears 4 times, which is more than any other shoe size.

Therefore, the mode is

7. Example 4 (Mean - Grouped Data): A survey was conducted on the number of hours students spent studying for an exam.

The results are shown below: | Hours | Frequency | | --------- | --------- | | 0 - 2 | 5 | | 2 - 4 | 8 | | 4 - 6 | 12 | | 6 - 8 | 7 | | 8 - 10 | 3 | Calculate the estimated mean study time.

Solution: | Hours | Frequency (f) | Midpoint (x) | f * x | | --------- | ------------- | ------------ | ----- | | 0 - 2 | 5 | 1 | 5 | | 2 - 4 | 8 | 3 | 24 | | 4 - 6 | 12 | 5 | 60 | | 6 - 8 | 7 | 7 | 49 | | 8 - 10 | 3 | 9 | 27 | | Total | 35 | | 165 | Mean = Σ(f * x) / Σf = 165 / 35 = 4.71 hours (approximately)

Example 5 (Median - Grouped Data): Using the same data as above, calculate the median study time. | Hours | Frequency (f) | Cumulative Frequency (CF) | | --------- | ------------- | ------------------------- | | 0 - 2 | 5 | 5 | | 2 - 4 | 8 | 13 | | 4 - 6 | 12 | 25 | | 6 - 8 | 7 | 32 | | 8 - 10 | 3 | 35 | Total frequency (n) = 35. n/2 = 17.

5. The median class is the class containing the 17.5th value, which is the 4 - 6 class. L = 4 (lower boundary of the median class) CF = 13 (cumulative frequency of the class before the median class) f = 12 (frequency of the median class) w = 2 (class width) Median = L + [(n/2 - CF)/f] w = 4 + [(17.5 - 13)/12] 2 = 4 + (4.5/12) * 2 = 4 + 0.75 = 4.75 hours 2.2 Measures of Dispersion: Measures of dispersion describe the spread or variability of the data.

Range: The difference between the highest and lowest values. It's a simple measure but highly sensitive to outliers. 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 and is less sensitive to outliers than the range. IQR = Q3 - Q

1. To find Q1 and Q3, determine the median of the lower and upper halves of the data, respectively.

Semi-Interquartile Range: Half of the interquartile range. It gives a more precise measure of dispersion around the median. Semi-IQR = (Q3 - Q1) /

2. Variance: The average of the squared differences from the mean. It measures how far the data points are spread out from the mean. A higher variance indicates greater variability.