Module 2 Quiz: Descriptive Statistics & Data Analysis

1

Frequency Distribution

When creating a frequency distribution for tablet weight data from 100 tablets, how many class intervals should you use according to Sturges' Rule?

Sturges' Rule: k = 1 + 3.322 log₁₀(n)

A) 5 intervals

B) 7 intervals

C) 8 intervals

D) 10 intervals

Step-by-Step Solution

Understanding: Sturges' Rule helps determine optimal class intervals for frequency distributions to balance detail with clarity.

Formula Application: k = 1 + 3.322 log₁₀(100)

Calculation: log₁₀(100) = 2, so k = 1 + 3.322(2) = 1 + 6.644 = 7.644

Rounding: Round to nearest integer = 8 intervals

Pharmaceutical Context: For tablet weight analysis, 8 intervals provide adequate resolution to detect weight variation patterns while maintaining statistical power.

Excel: =1+3.322*LOG10(100) = 7.644 ≈ 8

2

Central Tendency

A batch of tablets shows the following potency values (%). Which measure of central tendency is most appropriate for regulatory reporting?

Scenario: Tablet Potency Analysis

Tablet	1	2	3	4	5	6	7	8	9	10
Potency (%)	98.2	99.1	100.3	99.8	101.2	99.5	100.1	98.9	99.7	100.0

A) Mean (arithmetic average)

B) Median (middle value)

C) Mode (most frequent value)

D) Geometric mean

Step-by-Step Analysis

Understanding: For pharmaceutical potency data, we need a measure that represents typical performance and is sensitive to all values.

Mean Calculation: Sum = 98.2+99.1+100.3+99.8+101.2+99.5+100.1+98.9+99.7+100.0 = 996.8

Result: Mean = 996.8/10 = 99.68%

Regulatory Requirement: USP requires arithmetic mean for content uniformity calculations and acceptance value (AV) determination.

Pharmaceutical Context: Mean is standard for batch release testing as it's used in statistical quality control and process capability calculations.

Excel: =AVERAGE(A1:J1) = 99.68%

3

Normal Distribution

Drug absorption follows a normal distribution with mean Cmax = 45 μg/mL and standard deviation = 8 μg/mL. What percentage of patients will have Cmax between 37-53 μg/mL?

Bioavailability Study Parameters

Population: μ = 45 μg/mL, σ = 8 μg/mL
Range of Interest: 37 to 53 μg/mL
Distribution: Normal (bell curve)

A) 68%

B) 75%

C) 95%

D) 99.7%

Step-by-Step Solution

Understanding: We need to find P(37 ≤ X ≤ 53) using the 68-95-99.7 rule.

Calculate Boundaries: Lower: μ - σ = 45 - 8 = 37, Upper: μ + σ = 45 + 8 = 53

Recognize Pattern: The range 37-53 represents μ ± 1σ (one standard deviation from mean)

Apply 68-95-99.7 Rule: 68% of data falls within μ ± 1σ

Clinical Interpretation: 68% of patients will have Cmax values within this therapeutic range, useful for dose optimization.

Excel: =NORM.DIST(53,45,8,TRUE)-NORM.DIST(37,45,8,TRUE) = 0.68

4

Dispersion Analysis

Two analytical methods show the following results. Which method has better precision?

Analytical Method Comparison

Method	Mean	Standard Deviation
HPLC	250.0 mg	2.5 mg
UV Spectroscopy	125.0 mg	1.8 mg

A) HPLC (lower absolute standard deviation)

B) UV Spectroscopy (lower coefficient of variation)

C) HPLC (lower coefficient of variation)

D) Both methods have equal precision

Step-by-Step Analysis

Understanding: Precision comparison requires coefficient of variation (CV%) when means differ significantly.

HPLC CV%: CV% = (SD/Mean) × 100 = (2.5/250.0) × 100 = 1.0%

UV CV%: CV% = (SD/Mean) × 100 = (1.8/125.0) × 100 = 1.44%

Comparison: HPLC (1.0%) < UV (1.44%), therefore HPLC has better precision

Analytical Significance: Lower CV% indicates more consistent results relative to the measurement level, critical for method validation.

Excel HPLC: =(2.5/250)*100 = 1.0%
Excel UV: =(1.8/125)*100 = 1.44%

5

Data Visualization

A histogram of dissolution times shows a long tail extending to the right. This distribution is:

Dissolution Profile Analysis

Observation: Most tablets dissolve within 15-25 minutes, but some tablets take up to 45 minutes
Shape: Peak at left, tail extending right
Clinical Impact: Delayed dissolution may affect bioavailability

A) Positively skewed (right-skewed)

B) Negatively skewed (left-skewed)

C) Normally distributed

D) Bimodal distribution

Step-by-Step Analysis

Understanding: Skewness describes the asymmetry of a distribution around its mean.

Identify Pattern: Peak on left + tail extending right = positive skew

Memory Aid: The tail points in the direction of skewness (tail right = positive/right skew)

Pharmaceutical Context: Right-skewed dissolution suggests most tablets perform well, but a few outliers have slow release

Quality Implication: May indicate process issues causing occasional slow-dissolving units requiring investigation

Excel: Use SKEW function to calculate skewness coefficient (positive value confirms right skew)

6

Position Measures

For tablet hardness data (in kp): 4.2, 4.5, 4.8, 5.1, 5.3, 5.6, 5.8, 6.1, 6.4, 6.7. What is the interquartile range (IQR)?

Tablet Hardness Quality Control

Sorted Data: 4.2, 4.5, 4.8, 5.1, 5.3, 5.6, 5.8, 6.1, 6.4, 6.7 kp
Sample Size: n = 10 tablets
Quality Range: IQR represents middle 50% of hardness values

A) 1.2 kp

B) 1.5 kp

C) 1.8 kp

D) 2.5 kp

Step-by-Step Calculation

Understanding: IQR = Q3 - Q1, representing the spread of the middle 50% of data.

Find Q1 Position: (n+1)/4 = (10+1)/4 = 2.75, so Q1 is between 2nd and 3rd values

Calculate Q1: Q1 = 4.5 + 0.75(4.8-4.5) = 4.5 + 0.225 = 4.725 kp

Find Q3 Position: 3(n+1)/4 = 3(11)/4 = 8.25, so Q3 is between 8th and 9th values

Calculate Q3: Q3 = 6.1 + 0.25(6.4-6.1) = 6.1 + 0.075 = 6.175 kp

Calculate IQR: IQR = Q3 - Q1 = 6.175 - 4.725 = 1.45 ≈ 1.5 kp

Excel: =QUARTILE.INC(range,3)-QUARTILE.INC(range,1) = 1.5

7

Standardization

A tablet weight of 520 mg is measured from a batch with mean = 500 mg and standard deviation = 15 mg. What is the z-score and interpretation?

Weight Variation Analysis

Individual Tablet: 520 mg
Batch Statistics: μ = 500 mg, σ = 15 mg
Quality Concern: Is this tablet within acceptable range?

A) z = 1.33; tablet is unusually heavy

B) z = 1.33; tablet weight is normal

C) z = -1.33; tablet is unusually light

D) z = 0.75; tablet weight is normal

Step-by-Step Z-Score Calculation

Understanding: Z-score shows how many standard deviations a value is from the mean.

Formula: z = (x - μ)/σ where x = observed value, μ = mean, σ = standard deviation

Substitution: z = (520 - 500)/15 = 20/15 = 1.33

Interpretation: z = 1.33 means tablet is 1.33 standard deviations above average

Quality Assessment: Since |z| < 2, this is within normal variation (not unusually heavy)

Probability: ~91% of tablets weigh less than this one

Excel: =(520-500)/15 = 1.33
Probability: =NORM.S.DIST(1.33,TRUE) = 0.91

8

Frequency Analysis

From the dissolution time frequency table below, what percentage of tablets dissolved in 20 minutes or less?

Dissolution Testing Results

Time Interval (min)	Frequency	Cumulative Frequency
5-10	8	8
10-15	22	30
15-20	35	65
20-25	20	85
25-30	15	100

A) 35%

B) 65%

C) 85%

D) 100%

Step-by-Step Analysis

Understanding: "20 minutes or less" means all intervals up to and including the 15-20 minute interval.

Identify Relevant Intervals: 5-10 min, 10-15 min, and 15-20 min

Read Cumulative Frequency: At 20 minutes, cumulative frequency = 65 tablets

Calculate Percentage: (65/100) × 100% = 65%

Pharmaceutical Interpretation: 65% of tablets meet the dissolution requirement of ≤20 minutes, indicating good formulation performance.

Excel: =65/100*100 = 65%
Or use COUNTIFS for conditional counting

9

Sampling Statistics

A sample of 25 tablets from a large batch shows a standard deviation of 5 mg. What is the standard error of the mean (SEM)?

Sampling Analysis

Sample Size: n = 25 tablets
Sample SD: s = 5 mg
Purpose: Estimate precision of sample mean as batch estimator

A) 0.2 mg

B) 1.0 mg

C) 1.25 mg

D) 5.0 mg

Step-by-Step SEM Calculation

Understanding: SEM measures how precisely the sample mean estimates the population mean.

Formula: SEM = s/√n where s = sample standard deviation, n = sample size

Substitution: SEM = 5/√25 = 5/5 = 1.0 mg

Interpretation: The sample mean has an uncertainty of ±1.0 mg when estimating batch mean

Quality Significance: Smaller SEM indicates more precise batch estimates; increasing sample size reduces SEM

Excel: =5/SQRT(25) = 1.0 mg
Or: =STDEV(range)/SQRT(COUNT(range))

10

Data Visualization

A box plot of tablet disintegration times shows the median closer to Q1 than to Q3, with a long whisker extending toward higher values. This indicates:

Disintegration Time Analysis

Box Plot Features:
• Median line closer to bottom of box (Q1)
• Upper whisker longer than lower whisker
• Few tablets take much longer to disintegrate

A) Normal distribution

B) Positively skewed distribution

C) Negatively skewed distribution

D) Bimodal distribution

Step-by-Step Box Plot Analysis

Understanding: Box plot shape reveals distribution characteristics through quartile positions.

Median Position: Median closer to Q1 indicates more data concentrated in lower values

Whisker Analysis: Longer upper whisker shows tail extending toward higher values

Distribution Type: This pattern indicates positive (right) skewness

Pharmaceutical Context: Most tablets disintegrate quickly, but a few outliers take longer, possibly due to coating variations or compression issues

Excel: Create box plot with Insert > Charts > Box and Whisker to visualize distribution shape

11

Frequency Analysis

In a content uniformity test of 30 tablets, 6 tablets were found in the 95-100% range. What is the relative frequency for this class?

Content Uniformity Testing

Total Sample: 30 tablets tested
Class Interval: 95-100% potency
Absolute Frequency: 6 tablets in this range

A) 0.20 or 20%

B) 0.25 or 25%

C) 6.0 or 600%

D) 30.0 or 3000%

Step-by-Step Relative Frequency Calculation

Understanding: Relative frequency shows the proportion of total observations in each class.

Formula: Relative Frequency = Class Frequency / Total Frequency

Calculation: Relative Frequency = 6/30 = 0.20

As Percentage: 0.20 × 100% = 20%

Interpretation: 20% of tablets have potency between 95-100%, useful for assessing batch uniformity against specifications

Excel: =6/30 = 0.20 or =6/30*100 = 20%

12

Data Analysis

Drug release kinetics data shows strong positive skewness. Which transformation would most likely normalize the distribution?

Release Kinetics Analysis

Current Data: Highly right-skewed release times
Problem: Non-normal distribution violates statistical test assumptions
Goal: Transform to approximately normal distribution

A) Square transformation (x²)

B) Logarithmic transformation (log x)

C) Reciprocal transformation (1/x)

D) Square root transformation (√x)

Step-by-Step Transformation Selection

Understanding: Positive skewness requires transformations that compress the right tail more than the left.

Transformation Effects: Log transformation is most effective for strong positive skewness

Why Log Works: Logarithm compresses large values more than small values, reducing right-tail dominance

Alternative Order: For increasing skewness: √x < log x < 1/x (increasing compression)

Pharmaceutical Application: Drug kinetics often follow log-normal distributions, making log transformation appropriate

Verification: After transformation, test normality using Q-Q plots or Shapiro-Wilk test

Excel: =LOG(original_data) or =LN(original_data)
Check result: Create histogram of transformed data

13

Quality Control

Using the IQR method, a value is considered an outlier if it falls outside which range?

Outlier Detection Criteria

Method: Interquartile Range (IQR) Rule
Given: Q1 = 45 mg, Q3 = 55 mg, IQR = 10 mg
Purpose: Identify tablets with unusual weight values

A) Outside Q1 - IQR to Q3 + IQR

B) Outside Q1 - 1.5×IQR to Q3 + 1.5×IQR

C) Outside Q1 - 2×IQR to Q3 + 2×IQR

D) Outside Q1 - 3×IQR to Q3 + 3×IQR

Step-by-Step IQR Outlier Rule

Understanding: IQR method uses 1.5 times the interquartile range as the outlier boundary.

Lower Boundary: Q1 - 1.5×IQR = 45 - 1.5(10) = 45 - 15 = 30 mg

Upper Boundary: Q3 + 1.5×IQR = 55 + 1.5(10) = 55 + 15 = 70 mg

Outlier Range: Values < 30 mg or > 70 mg are outliers

Quality Control: This identifies tablets requiring investigation for process deviations

Box Plot Connection: These boundaries define whisker endpoints in box plots

Excel: Lower = QUARTILE.INC(range,1)-1.5*(QUARTILE.INC(range,3)-QUARTILE.INC(range,1))
Upper = QUARTILE.INC(range,3)+1.5*(QUARTILE.INC(range,3)-QUARTILE.INC(range,1))

14

Excel Application

Which Excel function would you use to create a frequency distribution for tablet weight classes automatically?

Excel Data Analysis

Task: Count tablets in weight ranges: 490-495, 495-500, 500-505, 505-510 mg
Data: 100 individual tablet weights in column A
Goal: Automated frequency counting

A) COUNTIF function

B) FREQUENCY function

C) HISTOGRAM function

D) PIVOT TABLE function

Step-by-Step Excel Function Selection

Understanding: FREQUENCY function is specifically designed for creating frequency distributions with defined bins.

FREQUENCY Syntax: =FREQUENCY(data_array, bins_array)

Setup: Data in A1:A100, bins in C1:C4 (495, 500, 505, 510)

Implementation: Enter as array formula (Ctrl+Shift+Enter) in D1:D5

Result: Automatically counts observations in each interval

Advantage: Updates automatically when data changes, ideal for pharmaceutical QC

Excel: =FREQUENCY(A1:A100,C1:C4)
Alternative: Data Analysis Toolpak > Histogram

15

Clinical Application

Two batches of tablets show identical means but different standard deviations (Batch A: σ=2mg, Batch B: σ=8mg). From a pharmaceutical quality perspective, which statement is correct?

Batch Comparison Analysis

Batch A: Mean = 250 mg, SD = 2 mg
Batch B: Mean = 250 mg, SD = 8 mg
Question: Which batch shows better manufacturing control?

A) Both batches have equal quality since means are identical

B) Batch A shows better process control and consistency

C) Batch B is superior due to higher variability

D) Standard deviation is irrelevant for pharmaceutical quality

Step-by-Step Quality Assessment

Understanding: Standard deviation measures process consistency and control, critical for pharmaceutical quality.

Batch A Analysis: SD = 2 mg indicates tight control, most tablets close to target weight

Batch B Analysis: SD = 8 mg indicates poor control, wide weight variation

Regulatory Impact: Lower variability reduces risk of out-of-specification units

Process Capability: Batch A likely has higher Cpk value, indicating better manufacturing capability

Patient Safety: Consistent dosing is crucial for therapeutic effectiveness and safety

Process Capability: Cpk = min[(USL-μ)/(3σ), (μ-LSL)/(3σ)]
Higher Cpk = better process control

Statistical Data Treatment in Formulation Development

Module 2: Descriptive Statistics & Data Analysis

Module 2 Assessment Quiz

When creating a frequency distribution for tablet weight data from 100 tablets, how many class intervals should you use according to Sturges' Rule?

A batch of tablets shows the following potency values (%). Which measure of central tendency is most appropriate for regulatory reporting?

Drug absorption follows a normal distribution with mean Cmax = 45 μg/mL and standard deviation = 8 μg/mL. What percentage of patients will have Cmax between 37-53 μg/mL?

Two analytical methods show the following results. Which method has better precision?

A histogram of dissolution times shows a long tail extending to the right. This distribution is:

For tablet hardness data (in kp): 4.2, 4.5, 4.8, 5.1, 5.3, 5.6, 5.8, 6.1, 6.4, 6.7. What is the interquartile range (IQR)?

A tablet weight of 520 mg is measured from a batch with mean = 500 mg and standard deviation = 15 mg. What is the z-score and interpretation?

From the dissolution time frequency table below, what percentage of tablets dissolved in 20 minutes or less?

A sample of 25 tablets from a large batch shows a standard deviation of 5 mg. What is the standard error of the mean (SEM)?

A box plot of tablet disintegration times shows the median closer to Q1 than to Q3, with a long whisker extending toward higher values. This indicates:

In a content uniformity test of 30 tablets, 6 tablets were found in the 95-100% range. What is the relative frequency for this class?

Drug release kinetics data shows strong positive skewness. Which transformation would most likely normalize the distribution?

Using the IQR method, a value is considered an outlier if it falls outside which range?

Which Excel function would you use to create a frequency distribution for tablet weight classes automatically?

Two batches of tablets show identical means but different standard deviations (Batch A: σ=2mg, Batch B: σ=8mg). From a pharmaceutical quality perspective, which statement is correct?

Quiz Complete!