Statistics and probability are essential tools across all professional fields for making informed decisions based on data. The ability to organize, summarize, and interpret data enables professionals in healthcare, business, agriculture, hospitality, and other sectors to identify trends, assess risks, and optimize resources. This chapter focuses on effective data presentation methods and introduces concepts of continuous and discrete variables, as well as grouped and ungrouped data, to equip students with foundational skills needed in real-world professional environments.

4.1 Data Presentation

Data presentation is the process of organizing and displaying data in a clear and meaningful way to facilitate analysis and decision-making. In Kenyan workplaces such as county referral hospitals, banks, and agricultural cooperatives, presenting data accurately helps stakeholders understand performance, identify challenges, and plan interventions. This section explains different types of data and how to present them effectively.

4.1.1 Continuous variables

Continuous variables are numerical data that can take any value within a given range and are measurable rather than countable. They are common in many sectors, for example, measuring patient blood pressure in hospitals or daily sales revenue in retail businesses.

Meaning of continuous variables

• Definition: Continuous variables represent data that can assume any value within an interval or range, including decimals and fractions.
• Measurement: These variables are measured using instruments or devices, such as thermometers or weighing scales.
• Examples: Examples include height, weight, temperature, time, and distance.
• Infinite Possibilities: Between any two values, there are infinitely many possible values.
• Precision: The precision depends on the measuring instrument; more precise instruments yield more decimal places.

Characteristics of continuous variables

• Quantitative Nature: Continuous variables are numerical and allow for arithmetic operations.
• Represented on Number Lines: They can be plotted on number lines with intervals.
• Data Distribution: Often follow known distributions such as normal or uniform.
• Data Collection: Requires careful measurement to minimize errors.
• Data Analysis: Suitable for calculating mean, median, mode, variance, and standard deviation.

Other ways of presenting continuous data

• Box Plots: Show data spread, median, quartiles, and outliers.
• Scatter Plots: Display relationships between two continuous variables.
• Tables: Summarize data in intervals with frequencies and cumulative frequencies.
• Frequency Polygons: Join the midpoints of the tops of histogram bars to compare two or more distributions on one graph.
• Dot Plots: Place each measurement as a dot along a number line, which suits small samples of measured values.

Handling measurement errors in continuous data

• Calibration: Instruments should be calibrated regularly to ensure accuracy.
• Precision Reporting: Report data with appropriate significant figures.
• Repeat Measurements: Taking multiple measurements reduces random errors.
• Data Cleaning: Remove or correct obvious errors or outliers.
• Documentation: Keep records of measurement conditions for transparency.

4.1.1.1 Histogram

A histogram is a graph of continuous data in which the data are sorted into class intervals and each class is shown as a bar. The bars touch each other because the classes follow on without gaps, and the area of each bar represents the frequency of its class. Histograms show at a glance where the values cluster and how they are spread. Example: a bank in Nakuru recorded the waiting time of 40 customers and drew a histogram, which showed that most customers waited between 5 and 15 minutes.

Frequency table for the waiting times:

Procedure for drawing a histogram

1. Sort the data into class intervals of the same width and count the frequency of each class.
2. Draw a horizontal axis for the variable, marked with the class boundaries, and label it with the units.
3. Draw a vertical axis for the frequency, starting from zero, and choose a scale that fits the highest frequency.
4. Draw a bar over each class, with the base from the lower boundary to the upper boundary and the height equal to the frequency.
5. Make sure the bars touch, with no gaps, because the variable is continuous.
6. Give the graph a title and label both axes, then read off the tallest bar to identify the most common class.

Histogram with unequal class widths

When the class widths are not equal, the height of a bar must be the frequency density, so that the area of the bar still represents the frequency.

Formula: frequency density = frequency ÷ class width

Substitution: classes 0 to under 10 (frequency 8), 10 to under 20 (frequency 16), and 20 to under 40 (frequency 12)

Working: 8 ÷ 10 = 0.8, 16 ÷ 10 = 1.6, and 12 ÷ 20 = 0.6

Result: the bar heights are 0.8, 1.6, and 0.6 per unit of the variable.

4.1.1.2 Line

A line graph joins plotted points with straight lines to show how a continuous variable changes, most often over time. It is the best choice for showing trends, rises, falls, and seasonal patterns. Useful for showing trends over time, e.g., monthly water consumption in a hotel. Example: a hotel in Naivasha plotted its water consumption for twelve months and saw that use rose in the dry months of January and February, so it planned its water storage before the dry season.

Procedure for drawing a line graph

1. Put time (or the other independent variable) on the horizontal axis and the measured variable on the vertical axis.
2. Choose a scale for each axis that spreads the data across the page, and start the vertical axis at zero when comparing sizes.
3. Plot a point for each pair of values, such as month and litres used.
4. Join the points in order with straight lines, using a ruler.
5. If more than one series is shown, use a different line style for each and add a key.
6. Title the graph, label the axes with units, and describe the trend, such as steady growth, a peak, or a dip.

4.1.2 Discrete variable

Discrete variables represent countable data usually expressed in whole numbers. They are widely used in various Kenyan sectors, such as counting the number of patients admitted daily in a hospital or the number of laptops issued in a TVET college.

Meaning of discrete variables

• Definition: Discrete variables represent data that can only take specific distinct values, typically integers.
• Countable: The values are countable and often result from counting objects or events.
• Examples: Number of employees, number of vehicles, number of customers.
• No Intermediate Values: No values exist between two consecutive integers.
• Categorical Link: Sometimes linked to categories but still numerical.

Characteristics of discrete variables

• Whole Numbers: Values are non-fractional and exact counts.
• Finite or Countably Infinite: The number of possible values is limited or countably infinite.
• Non-Measurable: Cannot be measured like length or weight.
• Data Analysis: Suitable for mode and median calculations; mean is possible but less informative.
• Frequency Counts: Often summarized as frequency distributions or bar charts.

Other ways of presenting discrete data

• Frequency Tables: List values and their corresponding frequencies.
• Dot Plots: Display individual data points along a number line.
• Pareto Charts: Highlight the most frequent categories for prioritization.
• Tally Charts: Record counts as marks in groups of five while data are being collected.
• Pictograms: Use a small picture to stand for a fixed number of items, which helps with non-technical audiences.

Common uses of discrete data in Kenyan workplaces

• Inventory Management: Counting stock items in retail businesses.
• Patient Records: Recording number of cases treated in hospitals.
• Attendance Tracking: Number of students attending classes in schools.
• Vehicle Fleet: Counting operational vehicles in county government offices.
• Customer Transactions: Number of daily transactions processed in banks.

4.1.2.1 Bar graph

A bar graph represents the frequency of each discrete value or category with a separate bar. The bars have equal width and are separated by equal gaps, because the values are distinct and there is nothing between them. The height of each bar shows the count, so the bars can be compared quickly. Example: a clinic run by a mission hospital in Chogoria counted the patients seen from Monday to Friday as 12, 9, 15, 11, and 18, and a bar graph showed that Friday was the busiest day.

Procedure for drawing a bar graph

1. List the categories or values and their frequencies in a table.
2. Draw the horizontal axis for the categories and the vertical axis for the frequency, starting from zero.
3. Choose a scale for the vertical axis so that the largest frequency fits on the page.
4. Draw a bar for each category, all with the same width.
5. Leave equal gaps between the bars.
6. Label both axes, add a title, and write the frequency above each bar if exact values are needed.

4.1.2.2 Pie graph

A pie graph is a circle divided into sectors, where the angle of each sector is proportional to the frequency of its category. It shows the share each category has of the whole, so it is best for comparing parts of a total with few categories. Example: a SACCO in Kisii recorded 100 loans by purpose: agriculture 40, education 30, business 20, and housing 10, and a pie graph showed that agriculture took the largest share.

Worked example of finding sector angles

Formula: angle = (frequency ÷ total frequency) × 360°

Substitution for agriculture: angle = (40 ÷ 100) × 360°

Working: 0.4 × 360 = 144

Result: the agriculture sector is 144°. In the same way, education is 108°, business is 72°, and housing is 36°, and the four angles add up to 360°.

Procedure for drawing a pie graph

1. Find the total of all the frequencies.
2. Calculate the angle for each category using the formula above.
3. Check that the angles add up to 360°.
4. Draw a circle with a compass and mark the centre.
5. Draw a first radius, then use a protractor to measure and draw each sector in turn.
6. Label each sector with its category and add a title.

4.1.3 Grouped data

Grouped data is data that has been organized into classes or intervals, often used when dealing with large datasets. Grouping simplifies analysis and is common in sectors like agricultural cooperatives analyzing crop yields or in counties assessing population age groups.

Meaning and purpose of grouped data

• Definition: Grouped data is data sorted into classes or intervals with corresponding frequencies.
• Purpose: Simplifies large datasets for easier interpretation.
• Data Aggregation: Combines individual data points into groups.
• Facilitates Analysis: Enables calculation of averages and measures of dispersion.
• Visual Presentation: Easier to display in histograms and frequency polygons.

Methods of grouping data

• Determine Range: Calculate difference between highest and lowest values.
• Decide Number of Classes: Usually between 5 and 15 classes for clarity.
• Calculate Class Width: Divide range by number of classes.
• Set Class Limits: Establish lower and upper boundaries for each class.
• Tally Frequencies: Count data points within each class.

Other ways of presenting grouped data

• Frequency Distribution Tables: Show classes and their frequencies.
• Frequency Polygons: Line graphs connecting midpoints of classes.
• Class Boundaries: Used to avoid gaps between classes in graphs. For whole-number classes such as 10 to 19 and 20 to 29, the boundaries are 9.5, 19.5, and 29.5.
• Class Midpoints: The value halfway between the lower and upper boundaries, found as (lower boundary + upper boundary) ÷ 2, used as the class representative.
• Relative Frequencies: The frequency of each class divided by the total, which allows groups of different sizes to be compared.

Advantages and limitations of grouped data

• Advantages:
• Simplifies complex data for easier understanding.
• Facilitates identification of trends and patterns.
• Saves space when presenting large datasets.
• Enables computation of statistical measures.
• Useful for visual comparison across groups.
• Limitations:
• Loss of individual data details.
• Less precise than ungrouped data.
• Choice of class intervals can affect interpretation.
• Assumes uniform distribution within classes.
• Can mask outliers or anomalies.

4.1.3.1 Histogram

A histogram for grouped data shows each class as a bar whose width covers the class boundaries and whose height gives the frequency (or frequency density when the widths are unequal). For classes written with whole numbers, the bars are drawn from the lower class boundary to the upper class boundary, so that there are no gaps between neighbouring bars. Example: a dairy cooperative in Meru grouped the daily milk deliveries of 60 farmers into classes and drew a histogram, which showed that most farmers delivered between 10 and 29 litres.

Procedure for drawing a histogram of grouped data

1. Write the classes in a frequency table and find the class boundaries.
2. Mark the class boundaries on the horizontal axis at equal spacing.
3. Scale the vertical axis from zero up to a value just above the highest frequency.
4. Draw each bar from its lower boundary to its upper boundary with height equal to the frequency.
5. If the class widths differ, calculate the frequency density for each class and use it as the height.
6. Label the axes with the variable and the units, then give the graph a title.

4.1.3.2 Bar

A bar graph for grouped data is used when the groups are categories, such as age bands, departments, or branches, rather than a continuous scale. The bars have gaps between them, which is the visible difference from a histogram. Example: a county government office in Kakamega grouped its population by age bands of 0 to 14, 15 to 34, 35 to 59, and 60 and above, and used a bar graph to compare the size of each band.

Procedure for drawing a bar graph of grouped data

1. Write the group names and their frequencies in a table.
2. Draw the axes and scale the vertical axis from zero.
3. Draw one bar of equal width for each group.
4. Leave equal gaps between the bars.
5. Shade or label the bars so that they can be told apart.
6. Add axis labels and a title.

4.1.3.3 Cumulative frequency

Cumulative frequency is the running total of the frequencies, obtained by adding each class frequency to the total of the classes before it. The last value must equal the total number of observations. It is used to find how many values lie below a given limit. Example: a TVET college in Kisii grouped the marks of 50 students and used cumulative frequency to find how many scored below 50.

Procedure for finding cumulative frequencies

1. Write the classes in order from the lowest to the highest.
2. Copy the first frequency as the first cumulative frequency.
3. Add the second frequency to the first cumulative frequency to get the second.
4. Continue adding each new frequency to the previous cumulative total.
5. Check that the last cumulative frequency equals the total of all frequencies.
6. Note that each cumulative frequency belongs to the upper class boundary of its class, for example 12 students scored below 50.

4.1.3.4 ogive

An ogive, also called a cumulative frequency curve, is the graph of the cumulative frequencies of grouped data. It is plotted against the upper class boundaries and joined with a smooth S-shaped curve. It is used to estimate the number of values below any limit and to read values such as the median and quartiles. Example: using the marks table above, the college draws an ogive and reads the horizontal value at height 25, which is half of 50, to estimate a median mark of about 59.

Procedure for drawing an ogive

1. Complete the table of cumulative frequencies.
2. Take the upper class boundary of each class as the x value, for example 40, 50, 60, and so on.
3. Add a starting point at the lower boundary of the first class (30) with a cumulative frequency of 0.
4. Plot each point (upper boundary, cumulative frequency) on graph paper.
5. Join the points with a smooth curve, not with straight lines.
6. To read a value, draw a horizontal line from the chosen cumulative frequency across to the curve, then drop a vertical line to the horizontal axis.

Frequency polygon from a histogram

Some workplaces prefer a frequency polygon to a histogram for comparing two sets of data on the same axes. The points are plotted at the class midpoints and joined by straight lines, and the polygon is closed on the axis at both ends.

4.1.4 Ungrouped data

Ungrouped data refers to raw data collected and listed without classification or grouping. It is often the starting point for data analysis in environments such as TVET colleges tracking student scores or retail outlets recording daily sales.

Meaning and features of ungrouped data

• Definition: Data presented in its original form without any organization into classes.
• Raw Data: Represents individual observations or measurements.
• Exact Values: Each data point is recorded precisely.
• Data Size: Suitable for small to moderate-sized datasets.
• Initial Step: Basis for further grouping or summarizing.

Methods of presenting ungrouped data

• Simple Lists: Data values arranged in order or as collected.
• Frequency Tables: Count occurrences of each value.
• Stem-and-Leaf Plots: Display data distribution while preserving individual values.
• Dot Plots: Show individual data points on a number line.
• Ordered Arrays: Data arranged in ascending or descending order.

Advantages and disadvantages of ungrouped data

• Advantages:
• Retains complete original data detail.
• Allows thorough analysis of individual values.
• Useful when dataset is small.
• Enables calculation of exact statistics.
• Facilitates identification of outliers.
• Disadvantages:
• Difficult to interpret with large datasets.
• Time-consuming to analyze manually.
• Hard to detect overall trends quickly.
• No summarization leads to cluttered presentation.
• Not suitable for visual data display.

Steps for organizing ungrouped data

1. Collect Data: Gather raw data from observations or records.
2. Review Data: Check for completeness and accuracy.
3. Sort Data: Arrange values in ascending or descending order.
4. Create Frequency Table: Count repetitions of each value.
5. Visualize: Use dot plots or stem-and-leaf if suitable.
6. Analyze: Calculate measures like mean and median directly.

4.1.4.1 Line

A line graph of ungrouped data plots each recorded value as a point and joins the points in the order in which the values were collected. It shows how individual values rise and fall, so it suits records such as daily sales or weekly attendance. Example: a bakery in Embu recorded its daily sales for one week as Ksh 12,000, Ksh 9,500, Ksh 10,200, Ksh 11,800, Ksh 15,400, Ksh 21,000, and Ksh 18,600, and a line graph showed that sales climbed sharply towards the weekend.

Procedure for drawing a line graph of ungrouped data

1. List the values in the order in which they were recorded, together with the day, month, or case number.
2. Draw the horizontal axis for the order (for example the days) and the vertical axis for the values.
3. Choose a vertical scale that fits the lowest and highest values.
4. Plot a point for each value above its day.
5. Join the points in order with straight lines.
6. Label the axes with units, add a title, and describe the pattern, such as the peak on Saturday.

4.1.4.2 Cumulative frequency

For ungrouped data, cumulative frequency is the running total of the frequencies in a frequency table, taken value by value. It tells how many observations are less than or equal to a given value. Example: a TVET college in Eldoret scored a quiz out of 5 for 20 students and used the cumulative frequency to find that 15 students scored 3 or less.

Procedure for finding and plotting cumulative frequency

1. Sort the raw values and build a frequency table of each distinct value.
2. Copy the first frequency as the first cumulative frequency.
3. Add each following frequency to the previous cumulative frequency.
4. Check that the last cumulative frequency equals the total number of values (20 in the example).
5. Plot each score against its cumulative frequency on a graph.
6. Join the points to show how the count builds up, using steps for discrete scores or a line for a smooth view, and read off the number of values at or below any score.

Chapter Summary

This chapter explored various methods of data presentation, including graphical techniques for continuous variables such as histograms and line graphs, as well as discrete variables using bar and pie graphs. It further examined grouped and ungrouped data presentations, incorporating histograms, bar charts, cumulative frequency graphs, and ogives. Measures of central tendency were discussed with detailed methods for calculating mean, mode, and median for both grouped and ungrouped data sets. The chapter then covered measures of dispersion, focusing on standard deviation and variance, again distinguishing between grouped and ungrouped data. Probability concepts were introduced with scenarios involving selection with and without replacement. Probability distribution functions were analyzed, highlighting the binomial and Poisson distributions as models for discrete random variables. Finally, the chapter addressed the normal distribution, emphasizing its importance in statistical analysis and probability theory.