Engineering Mathematics II applies statistical and probability concepts essential for biomedical engineers working in Kenya. These tools assist in analyzing experimental data, quality control of medical devices, and reliability assessment of biomedical systems. Mastery of central tendency and dispersion measures enables professionals to interpret clinical measurements and device performance accurately. Probability concepts underpin risk assessment and decision-making in biomedical engineering projects.

1.1 Measures of central tendency mean, mode and probability and median

Central tendency measures summarize data sets by identifying a central or typical value. Biomedical engineers use these in analyzing patient data such as blood pressure readings, enzyme activity levels, or device output measurements. Understanding mean, median, mode, and probability supports decision-making in diagnostics and device calibration.

1.1.1 Mean

The mean or average is the sum of all data points divided by the number of points. It represents the central value of data and is sensitive to extreme values (outliers).

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

where \(x_i\) are data points and \(n\) is the number of data points.

Worked Examples

Example 1: A biomedical engineer records heart rates (in bpm) from 5 patients: 72, 78, 75, 80, 76. Find the mean heart rate.

Given: \(n=5\), \(x_1=72\), \(x_2=78\), \(x_3=75\), \(x_4=80\), \(x_5=76\)

$$\bar{x} = \frac{72 + 78 + 75 + 80 + 76}{5}$$$$= \frac{381}{5}$$$$= 76.2$$

Answer: 76.2 bpm

Example 2: A lab measures glucose concentration (mmol/L) in 7 samples: 4.5, 5.0, 4.8, 5.2, 4.9, 5.1, 4.7. Calculate the mean glucose level.

Given: \(n=7\), \(x_i\) as above

$$\bar{x} = \frac{4.5 + 5.0 + 4.8 + 5.2 + 4.9 + 5.1 + 4.7}{7}$$$$= \frac{34.2}{7}$$$$= 4.8857$$

Answer: 4.89 mmol/L

Example 3: A device outputs voltage readings (V): 3.3, 3.0, 3.5, 3.4, 3.6, 3.7, 3.2, 3.8. Find the mean voltage.

Given: \(n=8\), \(x_i\) as above

$$\bar{x} = \frac{3.3 + 3.0 + 3.5 + 3.4 + 3.6 + 3.7 + 3.2 + 3.8}{8}$$$$= \frac{27.5}{8}$$$$= 3.4375$$

Answer: 3.44 V

Example 4: A clinical trial measures recovery times (days) of 10 patients: 12, 15, 14, 16, 13, 15, 14, 17, 18, 14. Calculate the mean recovery time.

Given: \(n=10\), \(x_i\) as above

$$\bar{x} = \frac{12 + 15 + 14 + 16 + 13 + 15 + 14 + 17 + 18 + 14}{10}$$$$= \frac{148}{10}$$$$= 14.8$$

Answer: 14.8 days

Example 5: A sensor measures oxygen saturation (%) in 6 samples: 95, 97, 96, 94, 98, 99. Find the mean oxygen saturation.

Given: \(n=6\), \(x_i\) as above

$$\bar{x} = \frac{95 + 97 + 96 + 94 + 98 + 99}{6}$$$$= \frac{579}{6}$$$$= 96.5$$

Answer: 96.5%

1.1.2 Median

The median is the middle value in an ordered data set, dividing data into two equal halves. It is robust to outliers, useful in skewed biomedical data like recovery times or enzyme levels.

To find the median:

• Order the data from smallest to largest.
• If \(n\) is odd, median is the middle value.
• If \(n\) is even, median is the average of the two middle values.

Worked Examples

Example 1: Find the median of systolic blood pressure (mmHg): 120, 130, 125, 140, 135.

Given: \(n=5\), data unordered

Order data: 120, 125, 130, 135, 140

Since \(n\) is odd, median is middle value \(x_3 = 130\)

Answer: 130 mmHg

Example 2: Find the median of enzyme activity (units) in 6 samples: 50, 55, 52, 60, 58, 53.

Given: \(n=6\)

Order data: 50, 52, 53, 55, 58, 60

Median = average of \(x_3\) and \(x_4\) = \(\frac{53 + 55}{2}\)

$$= \frac{108}{2}$$$$= 54$$

Answer: 54 units

Example 3: Determine the median of ECG signal amplitudes (mV): 0.8, 0.7, 1.0, 0.9, 0.85, 0.95, 1.05.

Given: \(n=7\)

Order data: 0.7, 0.8, 0.85, 0.9, 0.95, 1.0, 1.05

Median is middle value \(x_4 = 0.9\)

Answer: 0.9 mV

Example 4: Find the median of respiration rates (breaths/min): 16, 18, 20, 15, 19, 17.

Given: \(n=6\)

Order data: 15, 16, 17, 18, 19, 20

Median = average of \(x_3\) and \(x_4\) = \(\frac{17 + 18}{2}\)

$$= \frac{35}{2}$$$$= 17.5$$

Answer: 17.5 breaths/min

Example 5: Find the median of temperature readings (°C): 37.2, 36.8, 37.0, 37.5, 36.9.

Given: \(n=5\)

Order data: 36.8, 36.9, 37.0, 37.2, 37.5

Median is \(x_3 = 37.0\)

Answer: 37.0 °C

1.1.3 Mode

The mode is the most frequently occurring value in a data set. It is useful in biomedical engineering when identifying the most common measurement or value, such as the most frequent blood type or dominant device error code.

Worked Examples

Example 1: Find the mode of pulse rates (bpm): 70, 72, 75, 72, 70, 72, 74.

Given data

Frequency of values: 70 (2 times), 72 (3 times), 74 (1 time), 75 (1 time)

Mode is 72 bpm (highest frequency)

Answer: 72 bpm

Example 2: Find the mode of blood glucose levels (mmol/L): 5.1, 5.3, 5.1, 5.0, 5.3, 5.3, 5.2.

Frequency: 5.1 (2), 5.3 (3), 5.0 (1), 5.2 (1)

Mode is 5.3 mmol/L

Answer: 5.3 mmol/L

Example 3: Find the mode of sensor error codes: 101, 102, 101, 103, 101, 102, 104.

Frequency: 101 (3), 102 (2), 103 (1), 104 (1)

Mode is 101

Answer: 101

Example 4: Find the mode of patient recovery days: 14, 15, 14, 16, 15, 14, 17.

Frequency: 14 (3), 15 (2), 16 (1), 17 (1)

Mode is 14 days

Answer: 14 days

Example 5: Find the mode of oxygen saturation levels (%): 96, 97, 95, 96, 97, 96, 98.

Frequency: 96 (3), 97 (2), 95 (1), 98 (1)

Mode is 96%

Answer: 96%

1.1.4 Probability

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). In biomedical engineering, probability models help assess device failure risks or patient outcome chances.

For an event \(A\),

$$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$$

Worked Examples

Example 1: A biomedical device has 5 components; 1 is defective. What is the probability of randomly selecting a defective component?

Given: 1 defective, 5 total

$$P(\text{defective}) = \frac{1}{5}$$

Answer: 0.2

Example 2: In a batch of 100 blood samples, 10 are infected. Probability of selecting an infected sample?

$$P = \frac{10}{100} = 0.1$$

Answer: 0.1

Example 3: A test has 95% sensitivity. Probability that a randomly tested patient with the disease tests positive?

Given: Sensitivity = 0.95

$$P(\text{positive}|\text{disease}) = 0.95$$

Answer: 0.95

Example 4: A patient has 3 possible diagnoses: A, B, C with probabilities 0.5, 0.3, and 0.2 respectively. Probability the diagnosis is not B?

$$P(\text{not B}) = 1 - 0.3 = 0.7$$

Answer: 0.7

Example 5: Probability of device failure is 0.02. Probability it will not fail in a single test?

$$P(\text{no failure}) = 1 - 0.02 = 0.98$$

Answer: 0.98

Chapter Summary

This chapter covered essential statistical concepts starting with measures of central tendency including mean, mode, median, and their relationship with probability. It then explored measures of dispersion, highlighting how data spread is quantified. The concepts of variance and standard deviation were introduced to further describe data variability. Probability was defined clearly along with its foundational laws that govern event outcomes. Calculations involving discrete and continuous random variables demonstrated practical applications of these principles. The chapter examined key probability distributions such as binomial, Poisson, and normal distributions, explaining their characteristics and uses. Finally, the mean, variance, and standard deviation of these distributions were calculated to show their significance, and the application of probability distributions in engineering contexts was emphasized.