Curriculum Alignment
- Strand
- 3.0 Statistics and Probability
- Sub-Strand
- 3.1 Statistics 1
- Specific Learning Outcomes
- Determine mean, mode and median of grouped and ungrouped data
Subsection 3.1.2 Ungrouped Data
Ungrouped data refers to raw data collected from an experiment, survey, or observation that has not been organized into categories, classes, or intervals.
It is simply a list of individual values without any summarization or grouping.
For example, if you record the ages of 10 people as: 12, 15, 14, 13, 16, 15, 14, 12, 13, 15 this is ungrouped data because each value is listed exactly as collected.
Subsubsection 3.1.2.1 Frequency Distribution Tables
Teacher Resource 3.1.14.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 3.1.3.
Work in groups
The school administration wants to understand how much pocket money learners receive each week in order to plan a budgeting workshop.
The weekly pocket expenses (in Ksh) of \(25\) randomly selected students are:
120, 150, 180, 200, 220, 250, 270, 290, 300, 320, 350, 370, 390, 400, 420, 450, 470, 480, 490, 500, 340, 230, 280, 410, 330
(a)
Construct an ungrouped frequency distribution table showing each amount and its frequency.
(b)
Identify the highest and lowest pocket money received.
(c)
Discuss your work with your fellow learners.
Key Takeaway 3.1.15.
A frequency distrribution table is a table that shows an event and how many times it happens.
Ungrouped Frequency Distribution: Lists each individual value and how many times it occurs. Best for small datasets with few unique values (Example: student test scores: 75 appears 3 times, 80 appears 5 times).
Tally marks help visually count occurrences before recording the frequency.
Example 3.1.16.
45, 50, 55, 50, 60, 70, 75, 80, 70, 55, 60, 65, 50, 55, 45, 60, 75, 80, 70, 50
Prepare ungrouped frequency distribution table for the dataset.
Example 3.1.18.
The marks scored by 20 students in a mathematics test are:
12, 15, 17, 15, 19, 21, 23, 17, 19, 25, 21, 23, 19, 17, 15, 23, 25, 21, 19, 23
Prepare an ungrouped frequency distribution table for the data.
Checkpoint 3.1.19.
Checkpoint 3.1.20.
Checkpoint 3.1.21.
Checkpoint 3.1.22.
Exercises Exercises
1.
Twenty five students in Grade 10 recorded their travel time to school in minutes as follows:
15, 8, 22, 40, 12, 25, 8, 20, 15, 20, 15, 8, 15, 40, 12, 22, 12, 20, 8, 19, 20, 14, 12, 8, 22
-
Construct an ungrouped frequency distribution table for the data.
2.
The costs (in Ksh.) of manufacturing equipment were recorded as follows:
1250, 1425, 3870, 1250, 4175, 2100, 1425, 2370, 1250, 4195, 1250, 3870, 4175, 4195, 1250, 2100, 3525, 2100, 4175, 3870, 2100, 1250, 4195, 2100,1425
-
Construct an ungrouped frequency distribution table for the data.
3.
The annual rainfall (in mm) recorded in a region was as follows:
625, 645, 780, 645, 720, 745, 780, 1000, 645, 835, 780, 880, 1000,1050, 1000, 1050, 975, 1000, 625, 1050, 745, 720, 880, 780, 625
-
Construct an ungrouped frequency distribution table for the data.
Subsubsection 3.1.2.2 Mean
Curriculum Alignment
- Strand
- 3.0 Statistics and Probability
- Sub-Strand
- 3.1 Statistics 1
- Specific Learning Outcomes
- Determine mean, mode and median of grouped and ungrouped data
Teacher Resource 3.1.26.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 3.1.4.
Work in groups
-
Measure the shoe size of students in your class and record the data as shown below:\(34,32, 33, 35,36, 38,28,...\)
-
Construct a frequency distribution table for the data.
-
Find the Mean of the data.
-
How did you calculate the mean? Discuss the steps you took to find the mean and how it helps to understand the data.
-
Share your findings with other groups
Key Takeaway 3.1.27.
-
MeanMean is the sum of all values divided by the total number of values. It is also known as Arithmetic Average.The mean is calculated as follows:\begin{align*} \textbf{Mean} = \amp \frac{ \textbf{βX}}{ \textbf{N}} \end{align*}Where X represents the values in the dataset and N is the total number of values.Frequency distribution table can be used to find the mean for the ungrouped data. Using the formula below:\begin{align*} \overline{\textbf{x}} = \amp \frac{ \textbf{βfx}}{ \textbf{βf}} \end{align*}Where;
-
x represents the values in the dataset
-
\(\overline{\textbf{x}}\) is the mean
-
βfx is the sum of products of x and f
-
βf is the Sum of frequencies
-
Example 3.1.28.
In a class of \(30\) students, the test scores of students are:
45, 67, 89, 56, 45, 78, 90, 67, 81, 73, 55, 62, 77, 84, 91, 69, 58, 72, 88, 95, 60, 75, 45, 67, 80, 92, 87, 79, 68, 55
Find the mean
Solution.
\begin{align*}
\textbf{Mean} = \amp \frac{ \textbf{βX}}{ \textbf{N}}
\end{align*}
Where;
\(+77+84+91+69+58+72+88+95+60+75+45+67\)
\(+80+92+87+79+68+55 = 2096\)
\begin{align*}
= \amp \frac{2096}{30}\\
= \amp 69.87
\end{align*}
Therefore, the mean is \(69.87\)
Example 3.1.29.
The frequency distribution table below shows marks of \(20\) students in a Grade 10 class.
| Marks (x) | Frequency (f) |
|---|---|
| \(2\) | \(3\) |
| \(4\) | \(2\) |
| \(6\) | \(4\) |
| \(7\) | \(3\) |
| \(9\) | \(5\) |
| \(11\) | \(2\) |
| \(12\) | \(1\) |
Find the mean of the marks.
Solution.
Mean is given by:
| Marks (x) | Frequency (f) | fx |
|---|---|---|
| \(2\) | \(3\) | \(6\) |
| \(4\) | \(2\) | \(8\) |
| \(6\) | \(4\) | \(24\) |
| \(7\) | \(3\) | \(21\) |
| \(9\) | \(5\) | \(45\) |
| \(11\) | \(2\) | \(22\) |
| \(12\) | \(1\) | \(12\) |
| \(Ξ£x = 51\) | \(Ξ£f = 20\) | \(Ξ£fx = 138\) |
\begin{align*}
\overline{\textbf{x}} = \amp \frac{ \textbf{βfx}}{ \textbf{βf}} \\
= \amp \frac{138}{20}\\
= \amp 6.9
\end{align*}
Exercises Exercises
1.
The number of books borrowed by students from a school library in a week is as follows:
\(3, 5, 2, 4, 6, 3, 5, 7, 4, 3, 6, 2, 4, 5, 3, 6\)
Find the Mean (Average) number of books borrowed.
2.
The following frequency distribution table represents volume of water (in liters) contained in different bottles:
| Volume of water (liters) | Number of bottles |
|---|---|
| \(34.5\) | \(3\) |
| \(35.8\) | \(4\) |
| \(37.2\) | \(2\) |
| \(39.0\) | \(3\) |
| \(40.4\) | \(3\) |
Find the mean volume of water in the bottles.
3.
A company records the monthly salaries (in KES) of its employees.
| Salary (KES) | Frequency |
|---|---|
| \(25,000\) | \(8\) |
| \(30,000\) | \(15\) |
| \(35,000\) | \(12\) |
| \(40,000\) | \(10\) |
| \(50,000\) | \(5\) |
Calculate the mean.
4.
A factory records the number of products manufactured in a week:
150, 130, 220, 135, 180, 140, 125, 250, 145, 230, 200, 205, 145, 190, 155, 210, 225, 240, 135, 165, 245, 170, 175, 185, 130, 190, 195, 160,170, 150, 160, 120, 200, 210, 220, 235, 180, 230, 240, 215
Answer.
| Number of Products (x) | Frequency (f) | fx |
|---|---|---|
| \(120\) | \(1\) | \(120\) |
| \(125\) | \(1\) | \(125\) |
| \(130\) | \(2\) | \(260\) |
| \(135\) | \(2\) | \(270\) |
| \(140\) | \(1\) | \(140\) |
| \(145\) | \(2\) | \(290\) |
| \(150\) | \(2\) | \(300\) |
| \(155\) | \(1\) | \(155\) |
| \(160\) | \(2\) | \(320\) |
| \(165\) | \(1\) | \(165\) |
| \(170\) | \(2\) | \(340\) |
| \(175\) | \(1\) | \(175\) |
| \(180\) | \(2\) | \(360\) |
| \(185\) | \(1\) | \(185\) |
| \(190\) | \(2\) | \(380\) |
| \(195\) | \(1\) | \(195\) |
| \(200\) | \(2\) | \(400\) |
| \(205\) | \(1\) | \(205\) |
| \(210\) | \(2\) | \(420\) |
| \(215\) | \(1\) | \(215\) |
| \(220\) | \(2\) | \(440\) |
| \(225\) | \(1\) | \(225\) |
| \(230\) | \(2\) | \(460\) |
| \(235\) | \(1\) | \(235\) |
| \(240\) | \(2\) | \(480\) |
| \(245\) | \(1\) | \(245\) |
| \(250\) | \(1\) | \(250\) |
| \(Ξ£x = 4995\) | \(Ξ£f = 40\) | \(Ξ£fx = 7355\) |
Mean: \(183.88\) products.
Subsubsection 3.1.2.3 Median
Curriculum Alignment
- Strand
- 3.0 Statistics and Probability
- Sub-Strand
- 3.1 Statistics 1
- Specific Learning Outcomes
- Determine mean, mode and median of grouped and ungrouped data
Teacher Resource 3.1.33.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 3.1.5.
Work in groups
The following are the marks obtained by 15 students in a mathematics test:
45, 78, 56, 90, 67, 82, 54, 88, 73, 60, 95, 80, 70, 85, 68
(a)
Arrange the marks in ascending order.
(b)
Identify the median mark.
(c)
What does the median tell us about the score of the group?
(d)
Share your work with the class.
Key Takeaway 3.1.34.
The median is the middle value of a data set.
To find median, data must first be arranged in ascending or descending order.
If the number of observations is odd, the median is the value at the middle position.
Example: For the data set 45, 56, 60, 67, 68, 70, 73, 78, 80, 82, 85, 88, 90, 95, the median is \(73\) because it is the middle value when the data is arranged in order.
If the number of observations is even, the median is the average of the two middle values.
Example: For the data set 45, 56, 60, 67, 68, 70, 73, 78, the median is \((68 + 70) / 2 = 69\) because there are two middle values (68 and 70) when the data is arranged in order.
Example 3.1.35.
The ages of 6 students are: 12, 15, 14, 13, 16, 15. Find the median age.
Example 3.1.36.
The following data set represents the number of goals scored by a football team in 20 matches:
2, 3, 1, 4, 2, 5, 3, 2, 4, 1, 3, 2, 4, 5, 3, 2, 1, 4, 3, 2
Determine the median of the data set.
Solution.
First, we arrange the data in ascending order:
1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5
Since there are 20 observations (an even number), the median is the average of the two middle values, which are the 10th and 11th values in the ordered list.
The 10th value is 3 and the 11th value is also 3.
Median = \((3 + 3) / 2 = 3\text{.}\)
Example 3.1.37.
The weekly earnings (in Ksh) of 5 engineers in a company are as follows:
2000, 2500, 3000, 4500, 10000
Exercises Exercise
1.
Find the median of the following data:
50, 30, 10, 40, 20
2.
The number of goals scored in six football matches were:
2, 3, 1, 4, 2, 5
3.
The ages (in years) of students participating in a debate competition are:
14, 15, 14, 16, 15, 17, 16, 14, 15
Find the median age of the students.
4.
The daily earnings (in Ksh) of small business owners are:
500, 800, 750, 600, 900, 1200, 700
-
Find the median of the daily earnings.
5.
The following data set represents the number of hours spent on homework by a group of students in a week:
2, 3, 1, 4, 2, 5, 3, 4, 1, 2
Determine the median of the data set.
Subsubsection 3.1.2.4 Mode
Curriculum Alignment
- Strand
- 3.0 Statistics and Probability
- Sub-Strand
- 3.1 Statistics 1
- Specific Learning Outcomes
- Determine mean, mode and median of grouped and ungrouped data
Teacher Resource 3.1.38.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 3.1.6.
Work in groups
Your class is scheduling a favourable revision time. Students voted for their preferred time and the recorded votes are:
Afternoon, Evening, Afternoon, Morning, Afternoon, Evening, Morning, Afternoon, Evening, Morning, morning, Afternoon, Evening, Morning, Afternoon, Evening, Morning, Afternoon, Evening, Morning
(a)
Identify the mode
(b)
If the class chooses the most preferred time, when will the revision be held?
(c)
Explain why the mode is important in making this decision.
(d)
If the class had chosen the least preferred time, when would the revision have been held?
(e)
Share your work with the class
Key Takeaway 3.1.39.
The mode is the value that occurs most frequently in a data set.
A data set may have:
-
One mode (unimodal)
-
Two modes (bimodal)
-
More than two modes (multimodal)
-
No mode (if all values occur once)
The mode is useful in identifying the most common occurrence in real-life situations.
The mode does not require calculation β it requires counting and comparison.
Example 3.1.40.
The following data set represents the number of goals scored by a football team in 20 matches:
2, 3, 1, 4, 2, 5, 3, 2, 4, 1, 3, 2, 4, 5, 3, 2, 1, 4, 3, 2
Determine the mode of the data set.
Example 3.1.41.
The following data set represents the number of hours spent studying by a group of students in a week:
5, 6, 7, 5, 8, 6, 7, 5, 6, 8
Determine the mode of the data set.
Checkpoint 3.1.42.
Checkpoint 3.1.43.
Exercises Exercise
1.
The following data set represents the number of pets owned by a group of people:
0, 1, 2, 3, 1, 0, 2, 4, 1, 3
Determine the mode.
2.
The following data set represents the number of books read by a group of students in a month:
2, 3, 1, 4, 2, 5, 3, 2, 4, 1, 3, 2, 4, 5, 3
Determine the mode.
3.
The shoe sizes of students are:
38, 39, 40, 38, 41, 39, 38, 40, 39, 42
Find the common shoe size among the students.
4.
The mathematics test scores of Grade 10 students are:
75, 60, 85, 65, 90, 80, 50, 45, 70, 95
What is the mode of the test scores?
