By the end of this chapter, you will be able to: - accurately identify the difference between grouped and ungrouped data, - correctly organize ungrouped data using the right concepts, - clearly represent data in frequency tables for each category, - calculate the median for both grouped and ungrouped data following proper procedures, - accurately display data in chart form using standard methods, - confidently interpret information from charts using conventional techniques.
Mastering these skills will help you make sense of real-world data and communicate your findings effectively in the workplace.
Apply basic Statistics is an essential unit for Kenya TVET Diploma Level 6 students across all professional fields. Understanding how to identify and organize data is critical for making informed decisions in diverse sectors such as healthcare, education, banking, agriculture, and public administration. This chapter begins with the fundamental distinction between grouped and ungrouped data, providing a foundation for more advanced statistical methods used in professional environments.
In professional settings across Kenya, data is collected and presented in various forms. Being able to distinguish between grouped and ungrouped data allows practitioners to choose appropriate analysis techniques. For example, a county referral hospital managing patient records or a retail business analyzing daily sales figures must first identify the data type before proceeding with calculations.
Ungrouped data consists of raw data points collected directly from observations or measurements without any organization into groups or classes. This type of data is common in small datasets or when detailed individual data is required for analysis. Examples include the ages of students in a TVET college or daily rainfall measurements recorded at a weather station.
Ungrouped data is represented as a list or set of values, which can be discrete or continuous. The analysis of ungrouped data involves calculating measures such as mean, median, and mode from the raw values directly.
The general representation of ungrouped data is:
$$ x_1, x_2, x_3, \ldots, x_n $$
where \(x_i\) represents each individual data point and \(n\) is the total number of data points.
Example 1: A retail business records the number of customers visiting each day over 5 days as follows: 12, 15, 11, 14, 13 customers. Identify the type of data.
Given: Data points = 12, 15, 11, 14, 13 customers
Since the data is a list of individual counts without grouping:
Answer: The data is ungrouped data
Example 2: A SACCO records the monthly savings (in Ksh) of 7 members as: 1500, 2000, 1800, 2200, 1700, 1900, 2100. Identify the data type.
Given: Savings amounts = 1500, 2000, 1800, 2200, 1700, 1900, 2100 Ksh
This is a collection of individual savings values, not grouped.
Answer: The data is ungrouped data
Example 3: A county government office records the number of vehicles passing a checkpoint each hour for 8 hours: 30, 45, 40, 35, 50, 55, 60, 65. Identify the data type.
Given: Vehicle counts = 30, 45, 40, 35, 50, 55, 60, 65
This is raw data recorded hourly, not grouped.
Answer: The data is ungrouped data
Grouped data is data that has been organized into classes or intervals, often to simplify large datasets or to reveal patterns. This type of data is common in reports where detailed raw data is impractical to present, such as age groups of patients in a hospital or income ranges of cooperative members.
Grouped data is usually presented in frequency distribution tables showing class intervals and corresponding frequencies.
The general form of grouped data is:
| Class Interval | Frequency |
|---|---|
| \(a_1 - b_1\) | \(f_1\) |
| \(a_2 - b_2\) | \(f_2\) |
| \(\ldots\) | \(\ldots\) |
| \(a_k, b_k\) | \(f_k\) |
where \(a_i, b_i\) are the class limits and \(f_i\) the corresponding frequencies.
Example 1: A school records the marks of 30 students grouped as follows:
| Marks | Number of Students |
|---|---|
| 0–10 | 2 |
| 11–20 | 5 |
| 21–30 | 8 |
| 31–40 | 10 |
| 41–50 | 5 |
Identify the data type.
Given: Data presented in classes with frequencies.
Answer: The data is grouped data
Example 2: A hotel records the number of guests per room type over a month:
| Number of Guests | Frequency |
|---|---|
| 1–2 | 15 |
| 3–4 | 20 |
| 5–6 | 10 |
| 7–8 | 5 |
Given: Data summarized in intervals and frequencies.
Answer: The data is grouped data
Example 3: A farm cooperative records the weight of harvested maize in kilogram groups:
| Weight (kg) | Frequency |
|---|---|
| 0–50 | 3 |
| 51–100 | 7 |
| 101–150 | 12 |
| 151–200 | 8 |
Given: Weight classes with number of occurrences.
Answer: The data is grouped data
Distinguishing grouped from ungrouped data is vital for choosing suitable statistical methods. The key differences can be summarized as follows:
Example 1: A TVET college records the age of 6 students: 19, 20, 21, 22, 23, 24. Is this grouped or ungrouped data?
Given: Individual ages listed.
Answer: Ungrouped data
Example 2: A bank reports customer deposits in Ksh intervals:
| Deposit Range | Number of Customers |
|---|---|
| 0–10,000 | 50 |
| 10,001–20,000 | 30 |
| 20,001–30,000 | 20 |
Given: Data in intervals.
Answer: Grouped data
Example 3: A county referral hospital records daily patient visits for 7 days: 45, 50, 48, 52, 47, 49, 51. Identify the data type.
Given: Individual daily counts.
Answer: Ungrouped data
In practice, large ungrouped datasets are often converted into grouped data to facilitate analysis and reporting. This involves creating class intervals and tallying frequencies.
Steps for converting ungrouped data to grouped data:
Example 1: A retail business records daily sales (in Ksh) over 12 days: 1200, 1500, 1300, 1600, 1700, 1400, 1100, 1800, 1900, 1700, 1600, 1500. Convert to grouped data with 4 classes.
Given: Sales data and desired classes = 4
Step 1: Range = 1900 - 1100 = 800 Ksh
Step 2: Number of classes = 4
Step 3: Class width = 800 ÷ 4 = 200 Ksh
Step 4: Class limits:
Step 5: Tally frequencies:
Step 6: Frequency table:
| Sales (Ksh) | Frequency |
|---|---|
| 1100–1299 | 2 |
| 1300–1499 | 3 |
| 1500–1699 | 4 |
| 1700–1899 | 3 |
Answer: Data successfully grouped into 4 classes
Example 2: A cooperative records weights of 15 bags of maize (kg): 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115. Group into 5 classes.
Given: Data and classes = 5
Step 1: Range = 115 - 45 = 70 kg
Step 2: Number of classes = 5
Step 3: Class width = 70 ÷ 5 = 14 kg (round up to 15)
Step 4: Class limits:
Step 5: Tally frequencies:
Step 6: Frequency table:
| Weight (kg) | Frequency |
|---|---|
| 45–59 | 3 |
| 60–74 | 3 |
| 75–89 | 3 |
| 90–104 | 4 |
| 105–119 | 2 |
Answer: Data grouped into 5 classes
Example 3: A county government office records the number of cases handled monthly over 10 months: 23, 25, 28, 22, 27, 30, 31, 29, 26, 24. Group into 3 classes.
Given: Data and classes = 3
Step 1: Range = 31 - 22 = 9
Step 2: Number of classes = 3
Step 3: Class width = 9 ÷ 3 = 3
Step 4: Class limits:
Step 5: Tally frequencies:
Adjust class limits to include 31:
Re-tally:
Step 6: Frequency table:
| Cases | Frequency |
|---|---|
| 22–24 | 3 |
| 25–27 | 3 |
| 28–31 | 4 |
Answer: Data grouped into 3 classes with adjusted limits
A retail store records daily sales (Ksh) for 10 days as: 2500, 2700, 2600, 2800, 2900, 3000, 3100, 3200, 3300, 3400. Identify whether this is grouped or ungrouped data. (2 marks)
A hotel categorizes guest ages into the following groups with frequencies:
| Age (years) | Frequency |
|---|---|
| 18–25 | 20 |
| 26–33 | 25 |
| 34–41 | 15 |
| 42–49 | 10 |
Identify the data type. (2 marks)
Convert the following ungrouped data representing weights of coffee bags (kg): 45, 50, 55, 60, 65, 70, 75, 80 into 3 class intervals and create a frequency distribution table. (6 marks)
Given the following data for number of patients seen daily in a clinic: 12, 15, 10, 17, 14, 11, 13, group the data into 2 classes and tabulate the frequencies. (5 marks)
A cooperative society records monthly savings of members in Ksh as: 1000, 1500, 1200, 2000, 1700, 1600, 1400, 1800, 1900, 1300. Group the data into 4 classes and present the frequency table. (7 marks)
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Create a free accountThis chapter introduced the distinction between grouped and ungrouped data, emphasizing how each type is identified and handled. It then covered the organization of ungrouped data into a clear and systematic format to facilitate analysis. The construction of frequency tables was explained as an effective method for representing grouped data, allowing for easier interpretation. Methods to calculate the mean, mode, and median for both grouped and ungrouped data were demonstrated, showing how these measures of central tendency summarize data sets. Various types of charts were presented as visual tools to represent data, including their construction and appropriate use. Finally, the chapter explored how to interpret information from charts, enabling the extraction of meaningful insights from graphical data presentations. Together, these topics build a foundational understanding of basic statistics essential for analyzing and presenting data in practical contexts.
A county referral hospital records the number of patients attending its outpatient department over 7 days as follows: 45, 50, 38, 52, 48, 46, 49. Calculate the mean number of patients per day. (2 marks)
A retail business records daily sales (in Ksh 000) for 6 days as: 120, 135, 110, 115, 130, 125. Find the mode of the sales figures. (2 marks)
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