Applying statistical methods is essential for Science Laboratory Technology professionals in Kenya to analyze experimental data accurately and make informed decisions. This chapter focuses on processing raw data and understanding key statistical measures that summarize and describe data characteristics. Mastery of these concepts ensures reliable interpretation of laboratory results, quality control, and effective communication of findings.
In laboratory science, raw data collected from experiments or observations require proper processing to extract meaningful information. Processing includes organizing data, calculating measures of central tendency to find typical values, and preparing the dataset for further analysis.
Measures of central tendency summarize a dataset by identifying a central or typical value. The three main measures are mean, median, and mode, each providing different insights depending on data type and distribution. These measures are crucial for laboratory data interpretation such as determining average concentrations, typical reaction times, or common measurement values.
The mean, or arithmetic average, is the sum of all data values divided by the number of values. It is sensitive to extreme values but widely used due to its simplicity and mathematical properties.
$$\bar{x} = \frac{\Sigma x_i}{n}$$
where \(\bar{x}\) is the mean, \(\Sigma x_i\) is the sum of all values, and \(n\) is the number of values.
Example 1: A lab technician records the pH values of five water samples: 6.8, 7.0, 6.9, 7.1, 7.2. Find the mean pH.
Given: \(x_1 = 6.8\), \(x_2 = 7.0\), \(x_3 = 6.9\), \(x_4 = 7.1\), \(x_5 = 7.2\), \(n=5\)
$$\bar{x} = \frac{6.8 + 7.0 + 6.9 + 7.1 + 7.2}{5}$$
$$= \frac{35.0}{5}$$
$$= 7.0$$
Answer: Mean pH = 7.0
Example 2: A scientist measures the concentration of a solution in mg/L: 12.5, 15.0, 14.0, 13.5, 16.0, 15.5. Find the mean concentration.
Given: \(x_1=12.5\), \(x_2=15.0\), \(x_3=14.0\), \(x_4=13.5\), \(x_5=16.0\), \(x_6=15.5\), \(n=6\)
$$\bar{x} = \frac{12.5+15.0+14.0+13.5+16.0+15.5}{6}$$
$$= \frac{86.5}{6}$$
$$= 14.42 \text{ mg/L}$$
Answer: Mean concentration = 14.42 mg/L
Example 3: A microbiologist records bacterial colony counts: 120, 135, 150, 165, 180, 195, 210. Calculate the mean colony count.
Given: \(x_1=120\), \(x_2=135\), \(x_3=150\), \(x_4=165\), \(x_5=180\), \(x_6=195\), \(x_7=210\), \(n=7\)
$$\bar{x} = \frac{120 + 135 + 150 + 165 + 180 + 195 + 210}{7}$$
$$= \frac{1155}{7}$$
$$= 165$$
Answer: Mean colony count = 165
Example 4: The Kenya Medical Research Institute (KEMRI) laboratory records the pH values of five water samples: 6.8, 7.0, 6.9, 7.1, 7.2. Find the mean pH.
Given: \(x_1 = 6.8\), \(x_2 = 7.0\), \(x_3 = 6.9\), \(x_4 = 7.1\), \(x_5 = 7.2\), \(n=5\)
$$ \bar{x} = \frac{6.8 + 7.0 + 6.9 + 7.1 + 7.2}{5} $$$$ = \frac{35.0}{5} $$$$ = 7.0 $$
Answer: Mean pH = 7.0 (KEMRI water samples)
Example 5: At Nairobi City Water and Sewerage Company, a scientist measures the concentration of a solution in mg/L: 12.5, 15.0, 14.0, 13.5, 16.0, 15.5. Find the mean concentration.
Given: \(x_1=12.5\), \(x_2=15.0\), \(x_3=14.0\), \(x_4=13.5\), \(x_5=16.0\), \(x_6=15.5\), \(n=6\)
$$ \bar{x} = \frac{12.5+15.0+14.0+13.5+16.0+15.5}{6} $$$$ = \frac{86.5}{6} $$$$ = 14.42 \text{ mg/L} $$
Answer: Mean concentration = 14.42 mg/L (Nairobi City Water and Sewerage Company)
Example 4: At Egerton University laboratory, the median of pH values: 6.8, 7.0, 6.9, 7.1, 7.2 is calculated.
Given: \(n=5\), sorted data: 6.8, 6.9, 7.0, 7.1, 7.2
Median position: \(\frac{5+1}{2} = 3\)
Median = 3rd value = 7.0
Answer: Median pH = 7.0 (Egerton University lab)
Example 4: At the National Environment Management Authority (NEMA) lab, the mode of pH values: 6.8, 7.0, 6.9, 7.1, 7.0 is found.
Given data: 6.8, 6.9, 7.0, 7.0, 7.1
Mode = 7.0 (appears twice)
Answer: Mode = 7.0 (NEMA laboratory)
Example 4: At Kenya Bureau of Standards (KEBS), find the range of pH values: 6.8, 7.0, 6.9, 7.1, 7.2.
Given: \(x_{\text{max}}=7.2\), \(x_{\text{min}}=6.8\)$$
\text{Range} = 7.2 - 6.8 = 0.4
$$
Answer: Range = 0.4 (KEBS water samples)
Example 4: At Moi University laboratory, calculate variance of pH values: 6.8, 7.0, 6.9, 7.1, 7.2.
Given: \(n=5\), \(\bar{x} = 7.0\)
Calculate squared deviations:
(6.8 - 7.0)^2 = 0.04
(7.0 - 7.0)^2 = 0
(6.9 - 7.0)^2 = 0.01
(7.1 - 7.0)^2 = 0.01
(7.2 - 7.0)^2 = 0.04
Sum: 0.10
Variance:$$
s^{2} = \frac{0.10}{5 - 1} = \frac{0.10}{4} = 0.025
$$
Answer: Variance = 0.025 (Moi University lab)
Example 4: At JKUAT laboratory, using variance \(s^{2} = 0.025\) from pH data.$$ s = \sqrt{0.025} = 0.1581 $$ Answer: Standard deviation = 0.1581 (JKUAT laboratory)
The median is the middle value when data are arranged in ascending order. It is less affected by outliers and useful for skewed data.
Example 1: Find the median of pH values: 6.8, 7.0, 6.9, 7.1, 7.2.
Given: \(n=5\), sorted data: 6.8, 6.9, 7.0, 7.1, 7.2
Median position: \(\frac{5+1}{2} = 3\)
Median = 3rd value = 7.0
Answer: Median pH = 7.0
Example 2: Find the median concentration of: 12.5, 15.0, 14.0, 13.5, 16.0, 15.5.
Sorted: 12.5, 13.5, 14.0, 15.0, 15.5, 16.0, \(n=6\)
Median positions: 3rd and 4th values = 14.0 and 15.0
$$\text{Median} = \frac{14.0 + 15.0}{2} = 14.5$$
Answer: Median concentration = 14.5 mg/L
Example 3: Find the median bacterial colony count: 120, 135, 150, 165, 180, 195, 210.
Sorted data already given, \(n=7\)
Median position: \(\frac{7+1}{2} = 4\)
Median = 4th value = 165
Answer: Median colony count = 165
The mode is the most frequently occurring value in a dataset. It is useful for categorical data or identifying the most common measurement.
Example 1: Find the mode of pH values: 6.8, 7.0, 6.9, 7.1, 7.0.
Given data: 6.8, 6.9, 7.0, 7.0, 7.1
Mode = 7.0 (appears twice)
Answer: Mode = 7.0
Example 2: A scientist measures concentrations (mg/L): 12.5, 15.0, 14.0, 15.0, 16.0, 15.5.
Mode = 15.0 (appears twice)
Answer: Mode = 15.0 mg/L
Example 3: Bacterial colony counts: 120, 135, 150, 135, 180, 135, 195.
Mode = 135 (appears three times)
Answer: Mode = 135
Calculate the mean of the following absorbance readings: 0.12, 0.15, 0.14, 0.16, 0.13. (3 marks)
Find the median for the following enzyme activity levels (units): 25, 30, 28, 35, 40, 32. (3 marks)
Determine the mode of the following test results: 5.6, 5.8, 5.6, 6.0, 5.7, 5.6. (3 marks)
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Create a free accountThis chapter focused on applying statistical methods to analyze raw data effectively. It began with processing raw data by exploring measures of central tendency, including the mean, mode, and median, which summarize data sets by identifying typical values. The chapter then examined measures of dispersion such as range, quartiles, variance, and standard deviation, which describe the spread and variability within data. Understanding these measures is essential for evaluating how data points differ from the average and from each other. Finally, the chapter emphasized the interpretation of processed data to draw meaningful conclusions and support decision-making. Mastery of these statistical tools enables accurate data analysis in scientific and technical contexts.
A laboratory technician records the following glucose concentration (mg/dL) readings from five samples: 85, 90, 88, 92, 95. Calculate the mean glucose concentration. (2 marks)
The following absorbance values were recorded during a spectrophotometry experiment: 0.45, 0.50, 0.48, 0.50, 0.52, 0.50, 0.49. Find the mode of the absorbance values. (2 marks)