Lesson Notes By Weeks and Term v5 - Grade 10
Statistics – Week 5 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Statistics – Week 5 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 10
Term: 3rd Term
Week: 5
Theme: General lesson support
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Statistics is a vital branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. In Grade 10, we build upon the foundational statistical concepts learned in previous grades. This week, we focus on measures of central tendency (mean, median, and mode) and measures of dispersion (range, percentiles, quartiles, interquartile range, semi-interquartile range, and the five-number summary) which are crucial for understanding and interpreting data sets.
2.1 Measures of Central Tendency Mean (Average): The sum of all the values in a dataset divided by the number of values.
For ungrouped data: Mean = (Sum of all values) / (Number of values). For grouped data (frequency tables), the mean is estimated using: Mean ≈ Σ(f x) / Σf, where 'f' is the frequency and 'x' is the midpoint of each class interval. Why?* The mean gives us a general idea of the "center" of the data. How?* Sum all the values, then divide by the total number of values. Be careful to use the correct formula for grouped vs. ungrouped data.
Median: The middle value in a dataset 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. Why?* The median is less affected by outliers than the mean. How?* Arrange the data in order. Find the middle value. If there are two middle values, average them.
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 more than one mode (bimodal, trimodal, etc.). For grouped data, the modal class is the class with the highest frequency. Why?* The mode tells us which value is most common. How?* Count the occurrences of each value. The value that appears most often is the mode.
Example 1 (Ungrouped Data): Consider the ages of 7 Grade 10 learners: 15, 16, 15, 15, 17, 16,
1
5. Mean: (15 + 16 + 15 + 15 + 17 + 16 + 15) / 7 = 109 / 7 ≈ 15.57 years Median: Arrange in order: 15, 15, 15, 15, 16, 16,
1
7. The middle value is
1
5. So, Median = 15 years.
Mode: The age 15 appears most often (4 times). So, Mode = 15 years.
Example 2 (Grouped Data): The table shows the heights of 30 learners in centimetres: | Height (cm) | Frequency | |-------------|-----------| | 150 - 155 | 5 | | 155 - 160 | 8 | | 160 - 165 | 10 | | 165 - 170 | 7 | To estimate the mean: Find the midpoints of each class interval: 152.5, 157.5, 162.5, 167.5 Multiply each midpoint by its frequency: (152.5 5) + (157.5 8) + (162.5 10) + (167.5 * 7) = 762.5 + 1260 + 1625 + 1172.5 = 4820 Divide by the total frequency: 4820 / 30 ≈ 160.67 cm The modal class is 160-165 cm, because it has the highest frequency of 10. 2.2 Measures of Dispersion Measures of dispersion describe the spread or variability of data.
Range: The difference between the largest and smallest values in a dataset. Range = Maximum value – Minimum value. Why?* The range gives a simple measure of how spread out the data is. How?* Identify the largest and smallest values and subtract them.
Percentiles: Values that divide a dataset into 100 equal parts. The pth percentile is the value below which p% of the data falls.
Quartiles: Specific percentiles that divide the dataset into four equal parts.
Q1 (First Quartile): 25th percentile.
Q2 (Second Quartile): 50th percentile (which is also the median).
Q3 (Third Quartile): 75th percentile.
How to calculate Quartiles:* Arrange data in ascending order. Q2 is the median. Q1 is the median of the lower half of the data (excluding Q2 if n is odd). Q3 is the median of the upper half of the data (excluding Q2 if n is odd).
Interquartile Range (IQR): The difference between the third quartile (Q3) and the first quartile (Q1). IQR = Q3 – Q
1. Why?* The IQR measures the spread of the middle 50% of the data and is less affected by outliers than the range.
Semi-Interquartile Range: Half the interquartile range. SIQR = IQR / 2 = (Q3 – Q1) /
2. Why?* Provides a more precise measure of the spread around the median.
Five-Number Summary: A summary of the data consisting of the minimum value, Q1, median (Q2), Q3, and the maximum value. It is used to construct a box and whisker plot.
Box and Whisker Plot: A visual representation of the five-number summary. The box represents the IQR (Q1 to Q3), with a line marking the median. The whiskers extend from the box to the minimum and maximum values.
Example 3: Using the ages of 10 learners: 14, 15, 15, 16, 16, 16, 17, 17, 18,
1
8. Range: 18 - 14 = 4 years Median (Q2): (16 + 16) / 2 = 16 years Q1: Median of (14, 15, 15, 16, 16) = 15 years Q3: Median of (16, 17, 17, 18, 18) = 17 years IQR: 17 - 15 = 2 years Semi-Interquartile Range: 2 / 2 = 1 year Five-Number Summary: Minimum = 14, Q1 = 15, Median = 16, Q3 = 17, Maximum = 18 2.3 Outliers An outlier is a data point that is significantly different from other data points in the dataset. One common way to identify outliers is to use the following rule: Lower Bound = Q1 - 1.5 IQR Upper Bound = Q3 + 1.5 IQR Any value below the lower bound or above the upper bound is considered an outlier.
Example 4: Using data set: 10, 12, 14, 15, 16, 18, 20, 22, 50 Q1 =
1
2. IQR = 20-12=8; Q3=20 Lower bound = 12 - 1.5(8) = 0 Upper bound = 20 + 1.5(8) = 32 Therefore, 50 is considered an outlier. Guided Practice (With Solutions)
Question 1: Given the following data representing the number of hours students spend studying per week: 5, 7, 8, 6, 10, 4, 7, 9,
7. Calculate the mean, median, and mode.