This chapter focuses on applying statistical methods essential for Science Laboratory Technology professionals in Kenya. Accurate data collection and handling enable laboratory technologists to analyze experimental results, maintain quality control, and interpret findings reliably. Mastery of statistical data collection techniques ensures that laboratory outcomes are valid, reproducible, and meet regulatory standards. The following sections address the collection of raw data, distinguishing between ungrouped and grouped data formats relevant to laboratory contexts.

5.1 Collection of raw data

In laboratory science, raw data collection is the foundation for all subsequent analysis. Proper methods ensure data integrity and facilitate accurate interpretation of experiments, quality assessments, and research findings. Kenyan laboratory professionals frequently collect raw data from chemical assays, biological tests, and instrument readings, making statistical handling critical for decision-making and reporting.

5.1.1 Ungrouped data

Ungrouped data refers to raw data points collected individually without classification into categories or intervals. This data type is common in laboratory settings when measuring exact values such as pH levels, temperature readings, absorbance values, or counts of microbial colonies. Understanding ungrouped data allows technologists to compute basic statistics like mean, median, and mode directly from the raw observations.

Ungrouped data is represented as a simple list of observations, each corresponding to a single measurement or result. The focus is on handling individual values rather than summarizing them into frequency distributions. This approach is suitable when dealing with limited datasets or when detailed individual measurement analysis is required.

Mean of ungrouped data

The mean (average) is a measure of central tendency calculated by summing all observations and dividing by the number of observations. It provides an estimate of the typical value in the dataset.

$$ \bar{x} = \frac{\Sigma x_i}{n} $$

where \( \bar{x} \) is the mean, \( x_i \) are the individual data points, and \( n \) is the number of data points.

Worked Examples

Example 1: A laboratory records the pH of a water sample five times: 6.8, 7.0, 7.2, 6.9, and 7.1. Calculate the mean pH.

Given: \( x_1=6.8, x_2=7.0, x_3=7.2, x_4=6.9, x_5=7.1 \), \( n=5 \)

$$ \bar{x} = \frac{6.8 + 7.0 + 7.2 + 6.9 + 7.1}{5} $$

$$ \bar{x} = \frac{35.0}{5} $$

$$ \bar{x} = 7.0 $$

Answer: Mean pH = 7.0

Example 2: A technician measures the temperature (°C) of a chemical reaction at six intervals: 25, 27, 26, 28, 29, 27. Find the average temperature.

Given: \( x_i = 25, 27, 26, 28, 29, 27 \), \( n=6 \)

$$ \bar{x} = \frac{25 + 27 + 26 + 28 + 29 + 27}{6} $$

$$ \bar{x} = \frac{162}{6} $$

$$ \bar{x} = 27.0 $$

Answer: Mean temperature = 27.0 °C

Example 3: During a microbial assay, colony counts at five different plates are 120, 115, 125, 130, and 110 colonies. Calculate the mean colony count.

Given: \( x_i = 120, 115, 125, 130, 110 \), \( n=5 \)

$$ \bar{x} = \frac{120 + 115 + 125 + 130 + 110}{5} $$

$$ \bar{x} = \frac{600}{5} $$

$$ \bar{x} = 120 $$

Answer: Mean colony count = 120 colonies

Median of ungrouped data

The median is the middle value when the data points are arranged in ascending order. It divides the dataset into two equal halves and is less affected by extreme values.

To find the median:

1. Arrange data in ascending order.
2. If \( n \) is odd, median is the middle value.
3. If \( n \) is even, median is the average of the two middle values.

Worked Examples

Example 1: Find the median of pH values: 7.2, 6.8, 7.0, 7.1, 6.9.

Given: \( n=5 \) (odd)

Ordered data: 6.8, 6.9, 7.0, 7.1, 7.2

Median is the 3rd value (middle):

Answer: Median pH = 7.0

Example 2: Find the median temperature of readings: 25, 27, 26, 28, 29, 27.

Given: \( n=6 \) (even)

Ordered data: 25, 26, 27, 27, 28, 29

Median is average of 3rd and 4th values:

$$ \text{Median} = \frac{27 + 27}{2} $$

$$ \text{Median} = 27 $$

Answer: Median temperature = 27 °C

Example 3: Find the median of colony counts: 130, 115, 120, 125, 110.

Ordered data: 110, 115, 120, 125, 130

Median is the 3rd value:

Answer: Median colony count = 120 colonies

Mode of ungrouped data

The mode is the value that occurs most frequently in the dataset. It is useful in identifying the most common measurement or observation.

Worked Examples

Example 1: Determine the mode of pH readings: 6.8, 7.0, 7.0, 6.9, 7.1.

Values: 6.8 (1), 7.0 (2), 6.9 (1), 7.1 (1)

Answer: Mode = 7.0

Example 2: Find the mode of temperature readings: 25, 27, 26, 28, 29, 27.

Values: 25 (1), 27 (2), 26 (1), 28 (1), 29 (1)

Answer: Mode = 27 °C

Example 3: Find the mode of colony counts: 120, 115, 125, 130, 115.

Values: 115 (2), 120 (1), 125 (1), 130 (1)

Answer: Mode = 115 colonies

Range of ungrouped data

The range measures the spread of data by subtracting the smallest value from the largest value. It gives a simple indication of variability.

$$ R = x_{max} - x_{min} $$

Worked Examples

Example 1: Calculate the range of pH readings: 6.8, 7.0, 7.2, 6.9, 7.1.

$$ R = 7.2 - 6.8 = 0.4 $$

Answer: Range = 0.4 pH units

Example 2: Calculate the range of temperature readings: 25, 27, 26, 28, 29, 27.

$$ R = 29 - 25 = 4 $$

Answer: Range = 4 °C

Example 3: Calculate the range of colony counts: 120, 115, 125, 130, 110.

$$ R = 130 - 110 = 20 $$

Answer: Range = 20 colonies

5.1.2 Grouped data

Grouped data is organized into classes or intervals, commonly used when dealing with large datasets or continuous variables. Laboratory measurements such as temperature ranges, concentration intervals, or time durations are often grouped to simplify analysis and highlight data distribution.

Grouping data involves creating class intervals and tallying frequencies of observations within each interval. This format facilitates calculation of grouped mean, median, mode, and other statistics by summarizing data points into manageable groups.

Construction of frequency distribution tables

A frequency distribution table displays classes and the corresponding number of observations (frequency) in each class. It is constructed by dividing the range of data into intervals and counting how many data points fall into each.

Worked Examples

Example 1: A laboratory records the following temperature readings (°C): 25, 27, 26, 28, 29, 27, 30, 31, 32, 30. Construct a frequency distribution with class intervals of width 3 starting at 24.

Step 1: Determine classes

24–26, 27–29, 30–32

Step 2: Tally frequencies

• 24–26: 25, 26 → 2

• 27–29: 27, 28, 29, 27 → 4

• 30–32: 30, 31, 32, 30 → 4

Answer: Frequency distribution table as above

Example 2: Concentration readings (mg/L): 1.2, 1.5, 1.8, 2.0, 2.3, 2.5, 2.7, 3.0, 3.2, 3.5. Use class width 0.5 starting at 1.0.

Classes: 1.0–1.4, 1.5–1.9, 2.0–2.4, 2.5–2.9, 3.0–3.4, 3.5–3.9

Frequencies:

• 1.0–1.4: 1.2 → 1

• 1.5–1.9: 1.5, 1.8 → 2

• 2.0–2.4: 2.0, 2.3 → 2

• 2.5–2.9: 2.5, 2.7 → 2

• 3.0–3.4: 3.0, 3.2 → 2

• 3.5–3.9: 3.5 → 1

Answer: Frequency distribution table as above

Grouped mean calculation

The grouped mean estimates the average value from grouped data by using mid-class values weighted by frequencies.

$$ \bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} $$

where \( f_i \) is the frequency and \( x_i \) is the mid-class value of the \(i^{th}\) class.

Worked Examples

Example 1: Using temperature frequency table below, calculate the grouped mean:

Step 1: Calculate mid-class values:

• 24–26: \( \frac{24 + 26}{2} = 25 \)

• 27–29: \( \frac{27 + 29}{2} = 28 \)

• 30–32: \( \frac{30 + 32}{2} = 31 \)

Step 2: Calculate \( f_i x_i \):

• 2 × 25 = 50

• 4 × 28 = 112

• 4 × 31 = 124

Step 3: Sum frequencies and products:

$$ \Sigma f_i = 2 + 4 + 4 = 10 $$

$$ \Sigma f_i x_i = 50 + 112 + 124 = 286 $$

Step 4: Calculate mean:

$$ \bar{x} = \frac{286}{10} = 28.6 $$

Answer: Grouped mean temperature = 28.6 °C

Example 2: Calculate the grouped mean concentration from the frequency table:

Step 1: Mid-class values:

• 1.0–1.4: 1.2

• 1.5–1.9: 1.7

• 2.0–2.4: 2.2

• 2.5–2.9: 2.7

• 3.0–3.4: 3.2

• 3.5–3.9: 3.7

Step 2: Multiply frequencies by mid-class values:

• 1 × 1.2 = 1.2

• 2 × 1.7 = 3.4

• 2 × 2.2 = 4.4

• 2 × 2.7 = 5.4

• 2 × 3.2 = 6.4

• 1 × 3.7 = 3.7

Step 3: Sum frequencies and products:

$$ \Sigma f_i = 1 + 2 + 2 + 2 + 2 + 1 = 10 $$

$$ \Sigma f_i x_i = 1.2 + 3.4 + 4.4 + 5.4 + 6.4 + 3.7 = 24.5 $$

Step 4: Calculate mean:

$$ \bar{x} = \frac{24.5}{10} = 2.45 $$

Answer: Grouped mean concentration = 2.45 mg/L

Median for grouped data

The median class is the class where the cumulative frequency reaches or exceeds half of the total observations. The median is estimated by interpolation within this class.

$$ \text{Median} = L + ( \frac{\frac{N}{2} - F}{f} ) \times c $$

where

• \( L \) = lower boundary of median class

• \( N \) = total frequency

• \( F \) = cumulative frequency before median class

• \( f \) = frequency of median class

• \( c \) = class width

Worked Examples

Example 1: Using the temperature frequency table:

Total \( N=10 \), \( \frac{N}{2} = 5 \)

Median class: 27–29 (since cumulative frequency reaches 6)

Given:

• \( L = 27 \)

• \( F = 2 \) (cumulative frequency before median class)

• \( f = 4 \)

• \( c = 3 \)

Calculate median:

$$ \text{Median} = 27 + ( \frac{5 - 2}{4} ) \times 3 $$

$$ \text{Median} = 27 + ( \frac{3}{4} ) \times 3 $$

$$ \text{Median} = 27 + 2.25 = 29.25 $$

Answer: Median temperature = 29.25 °C

Example 2: Using concentration frequency table:

Total \( N=10 \), \( \frac{N}{2} = 5 \)

Median class: 2.0–2.4 (cumulative frequency 5)

Given:

• \( L = 2.0 \)

• \( F = 3 \)

• \( f = 2 \)

• \( c = 0.5 \)

Calculate median:

$$ \text{Median} = 2.0 + ( \frac{5 - 3}{2} ) \times 0.5 $$

$$ \text{Median} = 2.0 + (1) \times 0.5 $$

$$ \text{Median} = 2.5 $$

Answer: Median concentration = 2.5 mg/L

Mode for grouped data

The mode is estimated from the modal class, which has the highest frequency.

$$ \text{Mode} = L + ( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} ) \times c $$

where

• \( L \) = lower boundary of modal class

• \( f_1 \) = frequency of modal class

• \( f_0 \) = frequency of class before modal class

• \( f_2 \) = frequency of class after modal class

• \( c \) = class width

Worked Examples

Example 1: Using temperature frequency table:

Modal classes: 27–29 and 30–32 (both frequency 4). Choose 27–29 for calculation.

Given:

• \( L = 27 \)

• \( f_1 = 4 \)

• \( f_0 = 2 \)

• \( f_2 = 4 \)

• \( c = 3 \)

Calculate mode:

$$ \text{Mode} = 27 + ( \frac{4 - 2}{2 \times 4 - 2 - 4} ) \times 3 $$

$$ \text{Mode} = 27 + ( \frac{2}{8 - 6} ) \times 3 $$

$$ \text{Mode} = 27 + ( \frac{2}{2} ) \times 3 $$

$$ \text{Mode} = 27 + 3 = 30 $$

Answer: Mode temperature = 30 °C

Example 2: Using concentration frequency table:

Modal classes: 1.5–1.9, 2.0–2.4, 2.5–2.9, 3.0–3.4 (all frequency 2). Choose 1.5–1.9 for calculation.

Given:

• \( L = 1.5 \)

• \( f_1 = 2 \)

• \( f_0 = 1 \)

• \( f_2 = 2 \)

• \( c = 0.5 \)

Calculate mode:

$$ \text{Mode} = 1.5 + ( \frac{2 - 1}{2 \times 2 - 1 - 2} ) \times 0.5 $$

$$ \text{Mode} = 1.5 + ( \frac{1}{4 - 3} ) \times 0.5 $$

$$ \text{Mode} = 1.5 + (1) \times 0.5 $$

$$ \text{Mode} = 2.0 $$

Answer: Mode concentration = 2.0 mg/L

Chapter Summary

This chapter covered the application of statistical methods beginning with the collection of raw data, distinguishing between ungrouped data and grouped data. It then explored various techniques for data presentation including pictograms, histograms, pie charts, bar charts, and frequency polygons, each providing a visual way to interpret information. The chapter further addressed the processing of raw data to organize and simplify complex datasets. Measures of central tendency were examined in detail, focusing on the calculation and interpretation of the mean, mode, and median as key indicators of data distribution. Understanding these concepts equips students with the skills to analyze and summarize data effectively for scientific and practical applications. Through these topics, the chapter established a foundation for making informed decisions based on statistical evidence.