Skip to main content

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

Learner Experience 3.1.17.

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 3.1.18.

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 3.1.19.

The following data represents test scores of \(20\) students in a grade \(10\) class.
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.
Solution.
Table 3.1.20.
Test scores Tally Frequency
\(45\) \(//\) \(2\)
\(50\) \(////\) \(4\)
\(55\) \(///\) \(3\)
\(60\) \(////\) \(4\)
\(65\) \(/\) \(1\)
\(70\) \(///\) \(3\)
\(75\) \(//\) \(2\)
\(80\) \(//\) \(2\)

Example 3.1.21.

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.
Solution.
The ungrouped frequency distribution table for the given data is as follows:
Marks Tally Frequency
\(12\) \(/\) \(1\)
\(15\) \(///\) \(3\)
\(17\) \(///\) \(3\)
\(19\) \(/////\) \(5\)
\(21\) \(///\) \(3\)
\(23\) \(/////\) \(5\)
\(25\) \(//\) \(2\)

Checkpoint 3.1.22.

Checkpoint 3.1.23.

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
  1. Construct an ungrouped frequency distribution table for the data.
Answer.
Table 3.1.24.
Time (minutes) Frequency (f) Tally
\(8\) \(5\) \(/////\)
\(12\) \(4\) \(////\)
\(14\) \(1\) \(/\)
\(15\) \(4\) \(////\)
\(19\) \(1\) \(/\)
\(20\) \(4\) \(////\)
\(22\) \(3\) \(///\)
\(25\) \(1\) \(/\)
\(40\) \(2\) \(//\)

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
  1. Construct an ungrouped frequency distribution table for the data.
Answer.
Table 3.1.25.
Cost (Ksh.) Frequency (f) Tally
\(1250\) \(6\) \(//////\)
\(1425\) \(3\) \(///\)
\(2100\) \(5\) \(/////\)
\(2370\) \(1\) \(/\)
\(3525\) \(1\) \(/\)
\(3870\) \(3\) \(///\)
\(4175\) \(3\) \(///\)
\(4195\) \(3\) \(///\)

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
  1. Construct an ungrouped frequency distribution table for the data.
Answer.
Table 3.1.26.
Annual Rainfall (mm) Tally Frequency (f)
\(625\) \(///\) \(3\)
\(645\) \(///\) \(3\)
\(720\) \(//\) \(2\)
\(745\) \(//\) \(2\)
\(780\) \(////\) \(4\)
\(835\) \(/\) \(1\)
\(880\) \(//\) \(2\)
\(975\) \(/\) \(1\)
\(1000\) \(////\) \(4\)
\(1050\) \(///\) \(3\)

Subsubsection 3.1.2.2 Mean

Learner Experience 3.1.28.

Work in groups
  1. Measure the shoe size of students in your class and record the data as shown below:
    \(34,32, 33, 35,36, 38,28,...\)
  2. Construct a frequency distribution table for the data.
  3. Find the Mean of the data.
  4. How did you calculate the mean? Discuss the steps you took to find the mean and how it helps to understand the data.
  5. Share your findings with other groups

Key Takeaway 3.1.29.

  1. Mean
    Mean 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{∑X}}{ \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{∑fx}}{ \textbf{∑f}} \end{align*}
    Where;
    • x represents the values in the dataset
    • \(\overline{\textbf{x}}\) is the mean
    • ∑fx is the sum of products of x and f
    • ∑f is the Sum of frequencies

Example 3.1.30.

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{∑X}}{ \textbf{N}} \end{align*}
Where;
  • \(∑X\) is the sum of all values
  • \(N\) is the total number of values
\(∑X\) = \(45+67+89+56+45+78+90+67+81+73+55+62\)
\(+77+84+91+69+58+72+88+95+60+75+45+67\)
\(+80+92+87+79+68+55 = 2096\)
\begin{align*} = \amp \frac{2096}{30}\\ = \amp 69.87 \end{align*}
Therefore, the mean is \(69.87\)

Example 3.1.31.

The frequency distribution table below shows marks of \(20\) students in a Grade 10 class.
Table 3.1.32.
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
\(2\) \(3\) \(6\)
\(4\) \(2\) \(8\)
\(6\) \(4\) \(24\)
\(7\) \(3\) \(21\)
\(9\) \(5\) \(45\)
\(11\) \(2\) \(22\)
\(12\) \(1\) \(12\)
\(Σx = 51\) \(Σf = 20\) \(Σfx = 138\)
\begin{align*} \overline{\textbf{x}} = \amp \frac{ \textbf{∑fx}}{ \textbf{∑f}} \\ = \amp \frac{138}{20}\\ = \amp 6.9 \end{align*}

Checkpoint 3.1.33.

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.
Answer.
Mean (Average): 4 books.

2.

The following frequency distribution table represents volume of water (in liters) contained in different bottles:
Table 3.1.34.
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.
Answer.
Mean Volume: \(37.3\) liters.

3.

A company records the monthly salaries (in KES) of its employees.
Table 3.1.35.
Salary (KES) Frequency
\(25,000\) \(8\)
\(30,000\) \(15\)
\(35,000\) \(12\)
\(40,000\) \(10\)
\(50,000\) \(5\)
Calculate the mean.
Answer.
Mean: KES \(34,800\text{.}\)

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
  1. Make a frequency distribution table for the set of data.
  2. Calculate the mean.
Answer.
Number of Products (x) Frequency (f) fx
\(120\) \(1\) \(120\)
\(125\) \(1\) \(125\)
\(130\) \(2\) \(260\)
\(135\) \(2\) \(270\)
\(140\) \(1\) \(140\)
\(145\) \(2\) \(290\)
\(150\) \(2\) \(300\)
\(155\) \(1\) \(155\)
\(160\) \(2\) \(320\)
\(165\) \(1\) \(165\)
\(170\) \(2\) \(340\)
\(175\) \(1\) \(175\)
\(180\) \(2\) \(360\)
\(185\) \(1\) \(185\)
\(190\) \(2\) \(380\)
\(195\) \(1\) \(195\)
\(200\) \(2\) \(400\)
\(205\) \(1\) \(205\)
\(210\) \(2\) \(420\)
\(215\) \(1\) \(215\)
\(220\) \(2\) \(440\)
\(225\) \(1\) \(225\)
\(230\) \(2\) \(460\)
\(235\) \(1\) \(235\)
\(240\) \(2\) \(480\)
\(245\) \(1\) \(245\)
\(250\) \(1\) \(250\)
\(Σx = 4995\) \(Σf = 40\) \(Σfx = 7355\)
Mean: \(183.88\) products.

Subsubsection 3.1.2.3 Median

Learner Experience 3.1.37.

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.

(c)

What does the median tell us about the score of the group?

(d)

Share your work with the class.

Key Takeaway 3.1.38.

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 3.1.39.

The ages of 6 students are: 12, 15, 14, 13, 16, 15. Find the median age.
Solution.
First, we arrange the ages in ascending order: 12, 13, 14, 15, 15, 16.
Since there are 6 observations (an even number), the median is the average of the two middle values, which are 14 and 15.
Median = \((14 + 15) / 2 = 14.5\text{.}\)

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 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 3.1.41.

The weekly earnings (in Ksh) of 5 engineers in a company are as follows:
2000, 2500, 3000, 4500, 10000
Solution.
First, we arrange the earnings in ascending order: 2000, 2500, 3000, 4500, 10000.
Since there are 5 observations (an odd number), the median is the value at the middle position, which is the 3rd value in the ordered list.
Median = \(3000\) Ksh.

Checkpoint 3.1.42.

Checkpoint 3.1.43.

Exercises Exercise

1.

Find the median of the following data:
50, 30, 10, 40, 20
Answer.
Median = \(30\text{.}\)

2.

The number of goals scored in six football matches were:
2, 3, 1, 4, 2, 5
Answer.
Median = \((2 + 3) / 2 = 2.5\text{.}\)

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.
Answer.
Median = \(15\) years.

4.

The daily earnings (in Ksh) of small business owners are:
500, 800, 750, 600, 900, 1200, 700
  1. Find the median of the daily earnings.
  2. One business owner earned \(5000\) Ksh instead of \(1200\) Ksh. Calculate the new median.
Answer.
(a) Median = \(750\) Ksh.
(b) New median = \(800\) Ksh.

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.
Answer.
Median = \((2 + 3) / 2 = 2.5\) hours.

Subsubsection 3.1.2.4 Mode

Learner Experience 3.1.45.

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

(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 3.1.46.

The mode is the value that occurs most frequently in a data set.
A data set may have:
The mode is useful in identifying the most common occurrence in real-life situations.
The mode does not require calculation — it requires counting and comparison.

Example 3.1.47.

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.
Solution.
To find the mode, we count the frequency of each number of goals scored:
Number of Goals Frequency
1 3
2 6
3 5
4 4
5 2
The number of goals that occurs most frequently is 2, which occurs 6 times.

Example 3.1.48.

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.
Solution.
Using the frequency table, we count the frequency of each number of hours spent studying:
Hours Studied Frequency
5 3
6 3
7 2
8 2
The number of hours that occurs most frequently is 5 and 6, each occurring 3 times.

Checkpoint 3.1.49.

Checkpoint 3.1.50.

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.
Answer.
The number of pets that occurs most frequently is 1, which occurs 3 times.

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.
Answer.
The number of books that occurs most frequently is 2, which occurs 4 times.

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.
Answer.
The common shoe size among the students is 38 and 39, each occurring 3 times.

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?
Answer.
There is no mode since each score appears only once.
PrevTopNext