Data Accuracy and Integrity
Integrity- reliability – current and relevant
Accuracy – correctness
Some Factors which a ect data accuracy
1.
2.
3.
4.
5.
Bias
Lack of knowledge
Boredom – too much of data being
Rounding o
Intentional falsification
Action to maintain entry
1. Avoid manual entry
2. Checking and auditing
DATA CODING
1. Adding, Subtracting
Example: -95, -97, -98, -90
Add 100 to each: 5, 3, 2, 10
Coded mean: 5 ---- ((5+3+2+10)/4)
Uncoded Mean: 5-100 = -95
Standard deviation remains same and is not a ected by addition and subtraction.
S= 3.559
2. Multiplying or Dividing
Example: 1.05, 1.03, 1.02, 1.10
Multiply 100 to each: 105, 103, 102, 110
Coded Mean: 105
Un-coded mean: 105/100 =1.05
Standard deviation need to divided by you multiplied for coding.
For coded data S = 3.559
For Original data S = 3.559/100 = 0.03559
3. By truncation of repetitive terms
Example: 0.555, 0.553, 0.522, 0.550
Truncate 0.55 from all: 5, 3, 2, 0 (Multiplied by 1000 and subtracted 550 from each)
Coded mean: 2.5 --- ((5+3+2+0)/4)
Uncoded Mean : (2.5+550)/1000 = 0.525
Standard Deviation need to divide by you multiplied for coding
For coded data S: 2.0816
For original Data S: 2.0816/1000 = 0.0020816
Data Cleaning – Missing Data
In statistics, imputation is the process of replacing missing data with substituted values.
Missing data can introduce bias:
-
Missing Randomly
Reason for missing
Delete the row if the data is missing
Or Replace with the average value
Data Accuracy and Integrity
Factors of Data Quality Include
-
Integrity (Reliability – Current and Relevant)
Accuracy
Some factors which a ect data accuracy
-
Bias
Lack of knowledge
Boredom – too much of data being recorded
Rounding o
Intentional Falsification
Action to maintain data accuracy and integrity
-
Avoid manual entry
Checking and Auditing
Descriptive Statistics
Descriptive statistics refers to the process of summarizing and analysing data to describe its main features
in a clear and meaningful way. It is used to present raw data in a form that makes it easier to understand
and interpret. Descriptive statistics involves both graphical representations (such as charts and plots) and
numerical measures to summarize data e ectively. Unlike inferential statistics, which makes predictions
about a population based on a sample, Descriptive statistics is applied to data that is already known.
Inferential Statistics
Inferential statistics involves using data from a sample to make predictions, generalizations, or
conclusions about a larger population. Unlike descriptive statistics, which simply summarizes known data,
inferential statistics makes inferences or draws conclusions that go beyond the available data. It uses
probability theory to estimate population parameters and test hypotheses. By working with a
sample, inferential statistics allows researchers to make informed decisions without having to gather data
from an entire population.
Di erence between Descriptive and Inferential statistics
Descriptive Statistics
Inferential Statistics
It gives information about raw data which describes
the data in some manner.
It makes inferences about the population using data
drawn from the population.
It helps in organizing, analyzing, and to present data
in a meaningful manner.
It allows us to compare data, and make hypotheses and
predictions.
It is used to describe a situation.
It is used to explain the chance of occurrence of an
event.
It explains already known data and is limited to a
sample or population having a small size.
It attempts to reach the conclusion about the
population.
Examples include: mean, median, mode, range,
variance, histograms, pie charts.
Examples include: confidence intervals, hypothesis
testing, regression models, p-values.
Limited to presenting and analyzing known data.
Allows predictions and conclusions that go beyond the
data at hand.
Used for describing trends, organizing data for
presentation.
Used for predicting trends, testing hypotheses,
generalizing data from sample to population.
It can be achieved with the help of charts, graphs,
tables, etc.
It can be achieved by probability.
Types of Descriptive Statistics
1. Where the data centers (Measures of Central Tendency)
2. How spread out the data is (Measure of Variability)
3. How the data is distributed (Measures of Frequency Distribution)
1. Measures of Central Tendency
Statistical values that describe the central position within a dataset. There are three main measures of
central tendency:
Mean
Mean is the sum of all the components in a group or collection divided by the number of items in that
group or collection.
It is also defined as average which is the sum divided by count.
µ = ∑x/N
µ = population mean
xˉ = ∑x/ n
xˉ = Sample mean
Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 66 and 50. Determine the mean weight for the
provided collection of data.
Mean = Σx/n
= (54 + 32 + 45 + 61 + 20 + 66 + 50)/7
= 328 / 7
= 46.85
Thus, the group's mean weight is 46.85 kg.
Mode
Mode is one of the measures of central tendency, defined as the value that appears the most frequently in
the provided data, i.e. the observation with the highest frequency is known as the mode of data.
Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 45 and 50. Determine the mode weight for the
provided collection of data.
Mode = Most repeated observation in Dataset = 45
Thus, group's mode weight is 45 kg.
Median
Median of a data set is the value of the middle-most observation obtained after organizing the data in
ascending order, which is one of the measures of central tendency. Median formula may be used to
compute the median for many types of data, such as grouped and ungrouped data.
Ungrouped Data Median (n is odd): [(n + 1)/2]th term
Ungrouped Data Median (n is even): [(n / 2)th term + ((n / 2) + 1)th term]/2
where,

n = Number of Terms
Example: Weights of 7 girls in kg are 54, 32, 45, 61, 20, 66 and 50. Determine the median weight for the
provided collection of data.
Arrange the provided data collection in ascending order: 20, 32, 45, 50, 54, 61, 66
Median = [(n + 1) / 2]th term
= [(7 + 1) / 2]th term
= 4th term
= 50
Thus, group's median weight is 50 kg.
Percentile:
Measures of Dispersion
If the variability of data within an experiment must be established, absolute measures of variability
should be employed. These metrics often reflect di erences in a data collection in terms of the average
deviations of the observations. The most prevalent absolute measurements of deviation are mentioned
below.

Range

Standard Deviation

Variance
Range
The range represents the spread of your data from the lowest to the highest value in the distribution. It is
the most straightforward measure of variability to compute. To get the range, subtract the data set's
lowest and highest values.
Range = Highest Value – Lowest Value
Example: Calculate the range of the following data series: 5, 13, 32, 42, 15, 84
Arrange the provided data series in ascending order: 5, 13, 15, 32, 42, 84
Range = H - L
= 84 - 5
= 79
So, the range is 79.
Interquartile Range
Standard Deviation
S – for sample standard deviation
and xˉ is for sample mean
σ is for population standard deviation and µ is for population meam
Standard deviation (s or SD) represents the average level of variability in your dataset. It represents
the average deviation of each score from the mean. The higher the standard deviation, the more
varied the dataset is.
To calculate standard deviation, follow these six steps:
Step 1: Make a list of each score and calculate the mean.
Step 2: Calculate deviation from the mean, by subtracting the mean from each score.
Step 3: Square each of these di erences.
Step 4: Sum up all squared variances.
Step 5: Divide the total of squared variances by N-1.
Step 6: Find the square root of the number that you discovered.
Graphical Methods
Stem and Lef plot
A stem-and-leaf plot (or stemplot) is a simple way to display quantitative data while preserving the
original values. It organizes data into groups (stems) and shows the individual data points (leaves).
Structure of a Stem-and-Leaf Plot

Stem: The leading digit(s) of each data point (e.g., the tens place).

Leaf: The trailing digit(s) of each data point (e.g., the ones place).
Example:
Suppose we have the following test scores:
45, 52, 55, 57, 58, 64, 68, 72, 73, 81, 82, 85, 90
The stem-and-leaf plot would look like this:
4|5
5|2578
6|48
7|23
8|125
9|0
Steps to Make a Stem-and-Leaf Plot
1. Sort the data in ascending order.
2. Choose stems (usually the first digit or digits).
3. List leaves (the last digit) next to each stem.
4. Include a key to explain the interpretation.
Advantages

Retains original data values.

Shows distribution shape (like a histogram).

Easy to construct by hand.
Variations

Split Stemplot: Divides each stem into two (e.g., leaves 0-4 and 5-9).

Back-to-Back Stemplot: Compares two datasets side by side.
Box-and-whisker plot
A box-and-whisker plot (or box plot) is a graphical representation of a dataset that summarizes its
key statistical features, such as the median, quartiles, and potential outliers. It helps visualize the
distribution, spread, and skewness of the data.
Components of a Box-and-Whisker Plot
1. Minimum (Lower Whisker)
o
The smallest data point within 1.5 × IQR of the lower quartile (Q1).
o
Points below this may be outliers.
2. First Quartile (Q1)
o
The median of the lower half of the data (25th percentile).
o
The bottom edge of the box.
3. Median (Q2)
o
The middle value of the dataset (50th percentile).
o
The line inside the box.
4. Third Quartile (Q3)
o
The median of the upper half of the data (75th percentile).
o
The top edge of the box.
5. Maximum (Upper Whisker)
o
The largest data point within 1.5 × IQR of the upper quartile (Q3).
o
Points above this may be outliers.
6. Interquartile Range (IQR)
o
The range between Q1 and Q3 (IQR = Q3 – Q1).
o
The length of the box.
7. Outliers (if any)
o
Data points outside 1.5 × IQR from Q1 or Q3.
o
Often marked with dots or asterisks (*).
Example:
Dataset: 12, 15, 17, 18, 19, 20, 22, 23, 24, 25, 30, 35
1. Median (Q2) = Average of 20 and 22 → 21
2. Q1 (25th percentile) = Median of lower half (12,15,17,18,19,20) → 17.5
3. Q3 (75th percentile) = Median of upper half (22,23,24,25,30,35) → 24.5
4. IQR = Q3 – Q1 = 24.5 – 17.5 = 7
5. Lower Whisker = Smallest value ≥ Q1 – 1.5×IQR = 17.5 – 10.5 = 7 → 12 (no data below 12)
6. Upper Whisker = Largest value ≤ Q3 + 1.5×IQR = 24.5 + 10.5 = 35 → 30 (35 is an outlier)
Scatter Diagram (Scatter Plot)
A scatter diagram (or scatter plot) is a graphical tool used to visualize the relationship between two
numerical variables. It helps identify patterns, trends, and potential correlations between the
variables.
Key Features of a Scatter Plot
1. X-axis (Independent Variable) – The variable that is controlled or manipulated (e.g., study
hours).
2. Y-axis (Dependent Variable) – The variable that is measured or observed (e.g., exam score).
3. Data Points – Each dot represents a pair of (X, Y) values.
4. Trend Line (Optional) – A best-fit line (e.g., linear regression) to show the overall relationship.
5. Example: Study Hours vs. Exam Scores
Student
A
Study Hours (X)
2
Exam Score (Y)
50
B
4
60
C
D
6
8
75
85
Types of Relationships in a Scatter Plot
1. Positive Correlation (↑X → ↑Y)
o
Example: More study hours → Higher exam scores.
2. Negative Correlation (↑X → ↓Y)
o
Example: More screen time → Lower sleep quality.
3. No Correlation (No pattern)
o
Example: Shoe size vs. IQ (unrelated).
4. Non-Linear Relationship (Curved pattern)
o
Example: Age vs. Running speed (peaks at a certain age).
1. Uses of a Scatter Plot
✔ Identify trends and correlations.
✔ Detect outliers (unusual data points).
✔ Help in predictive modeling (e.g., regression analysis).
✔ Compare two variables across different groups.
Choosing the Right Method
Goal
Best Plot
Show distribution shape
Histogram / Density Plot
Compare groups
Box Plot / Bar Chart
Exact data values
Stem-and-Leaf / Dot Plot
Check for outliers
Box Plot
Compare proportions
Pie Chart / Bar Chart
Examine relationships
Scatter Plot
Track trends over time
Line Graph
GRAPHICAL METHODS FOR DEPICTING DISTRIBUTIONS
Errors of Statistical Tests


Type I error
Type II error
Type I Error (False Positive)

Definition: Rejecting a true null hypothesis (H0) when it is actually correct.

Also known as: False alarm, α-error.

Probability: Denoted by α (significance level).
o

Common α levels: 0.05 (5%), 0.01 (1%).
Example:
o
H0: A drug has no e ect.
o
If we conclude the drug works when it actually doesn’t, we commit a Type I
error.
2. Type II Error (False Negative)

Definition: Failing to reject a false null hypothesis (H0) when the alternative (H1) is
true.

Also known as: Missed detection, β-error.

Probability: Denoted by β.
o

Power of the test = 1−β1−β (probability of correctly rejecting H0 when H1 is
true).
Example:
o
H0: A drug has no e ect.
o

If we conclude the drug doesn’t work when it actually does, we commit
a Type II error.
Comparison of Type I and Type II Errors
Error
Type
Null Hypothesis
(H0)
Researcher's Decision
Type I (α)
True
Reject H0 (False Positive)
Type II (β)
False
Fail to reject H0 (False
Negative)
Consequence
Unnecessary action (e.g.,
approving ineffective drug)
Missed opportunity (e.g., failing to
detect a real effect)
Comparison Table: Type I vs. Type II Errors
Aspect
Type I Error (False Positive)
Type II Error (False Negative)
Definition
Rejecting a true H0
Failing to reject a false H0
Probability
α (e.g., 0.05)
β
Also Called
False alarm
Missed detection
Consequence
Unnecessary action
Missed opportunity
Control
Decrease α
Increase power (1 - β)
1. Type I Error (α): False alarm (claiming an e ect when none exists).
o
Controlled by setting a strict significance level (e.g., α = 0.01).
2. Type II Error (β): Missed detection (failing to find a real e ect).
o
Controlled by increasing sample size or e ect size.
3. Trade-o : Decreasing α increases β, and vice versa (unless sample size is
increased).
4. Power (1 - β): The probability of correctly rejecting H0H0 when H1H1 is true.
Confidence Level vs. Significance Level (Simple Explanation)
1. Confidence Level

What it means: How sure you are that your results are correct.

Example: A 95% confidence level means:
o
If you repeated your experiment 100 times, you’d expect the true result to
fall within your calculated range 95 times.

Where you see it: Confidence intervals (e.g., "We are 95% confident the average
height is between 165cm and 175cm").

Formula:
Confidence Level=1−αConfidence Level=1−α
(where α is the significance level).
2. Significance Level (α)

What it means: The risk you’re willing to take of being wrong (making a Type I Error).

Example: α=0.05 (5%) means:
o

You accept a 5% chance of falsely rejecting the null hypothesis (e.g., saying
a drug works when it doesn’t).
Where you see it: Hypothesis testing (p-values).
o
If p-value < α, you reject the null hypothesis.
Key Di erences
Aspect
Confidence Level
Significance Level (α)
Definition
How sure you are your results are correct
Risk of a false alarm (Type I Error)
Common Value
95% (or 90%, 99%)
0.05 (5%) or 0.01 (1%)
Used in
Confidence Intervals
Hypothesis Testing (p-values)
Relationship
Confidence Level=1−α
α=1−Confidence Level

Higher Confidence (e.g., 99%) → Lower αα (0.01) → Fewer false alarms, but harder to
detect real e ects (higher Type II Error risk).

Lower Confidence (e.g., 90%) → Higher αα (0.10) → Easier to detect e ects, but more
false alarms.
TYPE I ERROR WHEN SIGNIFICANCE LEVEL (Α) INCREASES
As the significance level (α) increases, the probability of a Type I Error (false positive) also
increases.
Detailed Explanation
1. Significance Level (α) = Max Allowable Type I Error Risk
o
By definition, α is the threshold you set for rejecting the null hypothesis.
o
Example: If α = 0.05 (5%), you accept a 5% chance of falsely rejecting H0
(Type I Error).
2. If You Increase α (e.g., from 0.05 to 0.10):
o
You’re allowing more false positives.
o
Type I Error rate goes up (from 5% to 10% in this case).
3. Why?
o
A higher α means you need less evidence to reject H0.
o
This makes it easier to claim "statistical significance", but also more likely to
be wrong when H0 is true.
Example: Drug Trial

Null Hypothesis (H0): "The new drug has no e ect."

α = 0.05: You have a 5% risk of falsely concluding the drug works.

α = 0.10: Now you have a 10% risk of the same false conclusion.
Trade-O with Type II Error

Increasing α reduces Type II Error (β):
o

You’re more likely to detect true e ects (higher power).
But at the cost of more false alarms (Type I Errors).
Key Takeaways
Action
E ect on Type I Error
E ect on Type II Error
Increase α (e.g., 0.05 → 0.10)
↑ Increases
↓ Decreases (better power)
Decrease α (e.g., 0.05 → 0.01)
↓ Decreases
↑ Increases (worse power)
When to Adjust α?

Use a lower α (e.g., 0.01) when false positives are costly (e.g., drug approvals).

Use a higher α (e.g., 0.10) when you prioritize detecting true e ects (e.g.,
exploratory research).
STATISTICAL POWER WHEN SAMPLE SIZE INCREASES
As sample size increases, statistical power also increases (you’re more likely to detect
a true e ect if it exists).
Why?
1. Statistical Power = Probability of Correctly Rejecting H0 (Avoiding Type II Error)
o
Power = 1−β, where β = Probability of Type II Error (false negative).
o
Higher power = Better chance of finding real e ects.
2. Larger Sample Size → More Precise Estimates
o
Reduces random variability (noise) in the data.
o
Makes it easier to distinguish a true signal (e ect) from background noise.
3. E ect on Hypothesis Testing
o
With more data, even small true e ects become statistically significant.
o
Example: A tiny improvement in a drug’s e ectiveness might only be
detectable with a large trial.
Example: Clinical Trial

Null Hypothesis (H0): "New drug = Placebo (no e ect)."

Truth: The drug does work (but the e ect is small).
o
Small sample (n=30): Might miss the e ect (low power → high β).
o
Large sample (n=1000): Likely detects the e ect (high power → low β).
Key Relationships
Factor
E ect on
Power
Why?
↑ Sample Size
↑ E ect Size
↓ Variability
↑ Increases
↑ Increases
↑ Increases
Less noise, clearer signal
Bigger e ects are easier to detect
Less scatter in data
↑ Significance Level
(α)
↑ Increases
More leniency to reject H0 (but raises Type I
Error risk)
Trade-O s

Cost vs. Power: Larger samples are expensive but reduce false negatives.

Diminishing Returns: Power increases fastest at small sample sizes; gains slow
down as n grows very large.
Practical Implications

Minimum Sample Size Calculation: Researchers often determine sample
size before a study to ensure adequate power (e.g., 80% power).

Underpowered Studies: Small samples risk missing real e ects (Type II Errors),
wasting resources.
Bottom Line: More data = More reliable conclusions. If you want to boost power,
increasing sample size is one of the most e ective ways!
Steps to Perform Hypothesis Testing
Hypothesis testing is a method used in statistics to make a decision about a population
based on a sample of data. Here's a step-by-step process:
Step 1: Define the Hypotheses

Null Hypothesis (H₀): This is a statement of no e ect or no di erence.

Alternative Hypothesis (H₁ or Ha): This is what you want to prove; it represents a
change or di erence.
Example:
H₀: μ = 100 (The mean is 100)
H₁: μ ≠ 100 (The mean is not 100)
Step 2: Set the Significance Level (α)

Common choices: 0.05, 0.01, or 0.10

This represents the probability of rejecting the null hypothesis when it is actually
true (Type I error).
Step 3: Choose the Right Test and Collect Data

Use Z-test, t-test, Chi-square, ANOVA, etc., depending on:
o
Sample size
o
Type of data
o
Known population standard deviation
Step 4: Calculate the Test Statistic
Step 5: Determine the p-value or Critical Value

p-value: Probability of obtaining a result as extreme as the one observed.

Critical value: The cuto point that defines the rejection region.
Step 6: Make the Decision

If p-value < α, reject H₀.

If using critical value:
o
If test statistic is in the rejection region, reject H₀.
Step 7: State the Conclusion

Clearly report whether there is enough evidence to support the alternative
hypothesis.
Example: One-Sample Z-Test
Problem:
A manufacturer claims that the mean weight of a product is 500g. A sample of 36 items
has a mean weight of 490g. The population standard deviation is known to be 20g. Test
at α = 0.05.
Step 1: Hypotheses

H₀: μ = 500

H₁: μ ≠ 500 (two-tailed test)
Step 2: Significance Level

α = 0.05
Step 3: Test to Use

Z-test (population SD known, n > 30)
Step 4: Test Statistic
Step 5: Critical Z-value for α = 0.05 (two-tailed)

Z₀.025 = ±1.96
Step 6: Decision

-3.00 < -1.96 → Reject H₀
Step 7: Conclusion

There is su icient evidence to say the mean weight is not 500g.
p- Value
The p-value is a key concept in hypothesis testing. It helps you determine whether your
observed data is statistically significant or could have happened by chance.
Definition:
The p-value is the probability of getting a result as extreme or more extreme than the
observed one, assuming the null hypothesis (H₀) is true.
Hypothesis Testing Basics:

H₀ (Null Hypothesis): The default assumption (e.g., there is no e ect, no di erence).

H₁ (Alternative Hypothesis): What you want to test/prove (e.g., there is an e ect,
there is a di erence).
How to Use the p-value:

Choose a significance level (α), commonly 0.05.

Calculate the p-value from your data.

Compare:
o
If p-value ≤ α → Reject H₀ → the result is statistically significant.
o
If p-value > α → Fail to reject H₀ → not statistically significant.
Example:
You test a new drug and want to see if it lowers blood pressure more than the current drug.

H₀: The new drug is no better than the current drug.

H₁: The new drug is better.
Your experiment gives a p-value of 0.02.

Since 0.02 < 0.05, you reject H₀ and conclude the new drug is statistically better.
Common Misunderstandings:

A low p-value does not prove the alternative hypothesis is true.

A high p-value does not prove the null hypothesis is true — it just means insu icient
evidence to reject it.

p-value ≠ probability that H₀ is true.
What is a p-value in hypothesis testing?
The p-value is a number that helps you determine whether the results of your experiment
or study are statistically significant. It tells you how likely it is to get the observed results
(or more extreme ones) if the null hypothesis (H₀) were true.
Key Concepts

Null Hypothesis (H₀): A statement that there is no e ect or no di erence.

Alternative Hypothesis (H₁ or Ha): A statement that there is an e ect or di erence.

Significance Level (α): The threshold you choose (commonly 0.05). If p-value ≤ α,
you reject H₀.
Interpretation of p-value
p-value What it means
Decision
p ≤ 0.05 The observed result is unlikely under H₀ Reject H₀ (evidence supports H₁)
p > 0.05 The observed result is likely under H₀
Fail to reject H₀
Example:
Suppose a drug company claims a new pill lowers blood pressure.

H₀: The pill has no e ect.

H₁: The pill does lower blood pressure.
You conduct a study and calculate a p-value = 0.03.

Since 0.03 < 0.05, you reject H₀.

Conclusion: There is statistical evidence that the pill works.
Common Misunderstandings:

p-value is NOT the probability that H₀ is true.

p-value is NOT the probability of making a mistake.

Low p-value ≠ practical importance (only statistical significance).
Sample Size
What is Sampling?
Sampling is the process of selecting a subset (sample) from a population to estimate
characteristics of the whole population.
We use sampling when:

It’s impractical or too costly to inspect the entire population.

We need to make inferences based on limited data.
Key Definitions
Term
Definition
Population Entire set of items or individuals of interest
Sample
A subset of the population selected for study
Parameter A numerical value that describes a population (e.g., population mean μ)
Statistic
A numerical value that describes a sample (e.g., sample mean ̄x)
Types of Sampling Methods
1. Probability Sampling (Randomized)
Every element has a known, non-zero chance of being selected.

Simple Random Sampling
Each item has an equal chance.
Example: Drawing names from a hat.

Systematic Sampling
Select every k-th item.
Example: Every 10th product on a conveyor belt.

Stratified Sampling
Divide population into subgroups (strata), then randomly sample from each.
Useful when subgroups di er significantly.

Cluster Sampling
Divide into clusters, randomly select clusters, then sample all items within.
Useful when population is spread geographically.
2. Non-Probability Sampling (Non-Random)
Not every item has a known chance of selection.

Judgmental Sampling
Based on expert opinion or experience.

Convenience Sampling
Based on ease of access. Less reliable for inference.
Sampling Distributions
A sampling distribution is the distribution of a statistic (like the sample mean) over many
samples from the same population.
According to the Central Limit Theorem, as sample size increases (n ≥ 30), the sampling
distribution of the sample mean tends to be normally distributed, even if the population
isn't.
Sampling Plans in Quality Engineering
In quality control, we often use acceptance sampling:
Types:

Attribute Sampling: Inspection is pass/fail (go/no-go).
E.g., “Accept if ≤ 2 defects in 50 items.”

Variable Sampling: Uses measurement data (like length or weight).
E.g., “Accept if sample mean is within tolerance.”
Terms:

AQL (Acceptable Quality Limit): The maximum defective level that’s considered
acceptable.

LTPD (Lot Tolerance Percent Defective): The quality level considered unacceptable.

Producer’s Risk (α): Risk of rejecting a good lot.

Consumer’s Risk (β): Risk of accepting a bad lot.
Example
You're tasked to determine whether a batch of syringes meets a design spec. Instead of
testing all 10,000 units, you randomly select 50.
You calculate:

Sample mean

Sample standard deviation

Confidence interval (to estimate the true mean)
You then decide whether to accept or reject the batch.
Sure! Here's a realistic example of sampling as it might appear in your CQE exam or realworld quality engineering role:
Example: Sampling for Syringe Diameter
Scenario:
You work at a medical device company manufacturing syringes. The specification for the
inner diameter of the syringe barrel is 10.00 mm ± 0.20 mm.
There are 10,000 syringes in a production lot. Measuring all of them is not practical, so you
decide to take a sample.
Step 1: Select a Sample

Sampling Method: Simple random sampling

Sample Size (n): 30 syringes

Measured Diameters (in mm):
9.82, 10.01, 10.15, 9.96, 10.08, 9.89, 10.12, 9.94, 9.87, 10.03,
10.05, 10.00, 9.95, 10.10, 10.02, 9.98, 10.06, 9.93, 9.91, 10.04,
9.97, 10.09, 10.00, 9.88, 9.92, 10.07, 10.11, 9.90, 9.99, 10.14
Step 4: Make a Decision
The specification limits are (9.80 mm, 10.20 mm).
Since the entire confidence interval lies within the spec, the batch is considered
acceptable.
CQE Takeaway

You used simple random sampling to inspect a small subset.

Calculated sample mean, standard deviation, and confidence interval.

Used statistical inference to decide lot acceptability.

Demonstrated understanding of variable data sampling.
Example: Inspecting Catheter Lengths
Scenario:
Your company produces catheters, and each catheter must be 150 mm ± 2 mm in length
(specification limits: 148 mm to 152 mm).
You take a sample of 50 catheters from a production lot of 5,000 units to check if the
process is under control.
Step 1: Collected Sample Data
You measured 50 catheters and found:

Sample Mean (x
̄ ) = 149.6 mm

Sample Standard Deviation (s) = 1.1 mm

Sample Size (n) = 50
Step 3: Compare with Specification Limits

Spec limits: (148.0 mm, 152.0 mm)

The entire confidence interval (149.295 to 149.905) is within the specification limits.
Conclusion:
The process is capable based on the sample, and the lot is likely acceptable. However, this
doesn’t guarantee every unit is in spec—it tells us the average and spread are acceptable
at the 95% confidence level.
Why This Is Important for CQE:

Understand when to use Z vs. t-distribution

Know how to interpret confidence intervals

Make inferences about population parameters based on samples

Evaluate process capability from sample data
Example: Proportion of Defective Syringes
Scenario:
You're auditing a production lot of 5,000 syringes. Each syringe is visually inspected for
surface cracks — a go/no-go check.
You randomly sample 100 syringes and find that 6 of them are defective.
Step 1: Calculate the Sample Proportion (p
̂ )
p^=xn=6100=0.06\hat{p} = \frac{x}{n} = \frac{6}{100} = 0.06
Step 3: Decision Based on AQL
Let’s assume the AQL (Acceptable Quality Level) is 5% (0.05).

The upper bound of the confidence interval is 10.65%, which exceeds the AQL.

This means we cannot be confident that the true defect rate is below the
acceptable level.
Conclusion:
Even though only 6 syringes were defective in the sample, the statistical analysis suggests
that the true population defect rate might exceed the AQL, so the lot should be rejected or
investigated further.
CQE Takeaways:

Use this method when dealing with attribute data (pass/fail).

The confidence interval for a proportion helps assess if the true defect rate is within
acceptable limits.

If the upper confidence bound > AQL → process not acceptable.
Calculating sample size for a proportion is a crucial, especially in attribute sampling.
CQE Tips:

If no prior estimate is available for p^\hat{p}, use 0.5 (most conservative estimate).

Always round up sample size to the next whole number.

Smaller margins of error → larger samples required.
When to use the t-distribution instead of the z-distribution is essential for both the CQE
exam and practical quality engineering.
Z vs. t Distribution: When to Use Which?
Condition
Use Z
Use t
Population standard deviation (σ) is
known
Yes
No
Sample standard deviation (s) is used (σ
unknown)
No
Yes
Yes (can
approximate Z)
Sample size is large (n ≥ 30)
Sample size is small (n < 30)
No
(but still t if σ is
unknown)
Must use t
Key Rule of Thumb:
Situation
Use
Small sample (n < 30) + σ unknown t-distribution
Large sample (n ≥ 30) + σ unknown You can approximate with Z, but t is still more correct
σ known (very rare in real life)
Z-distribution
Example Scenarios

You measure the diameter of 12 orthopedic implants, and population standard
deviation is unknown:
→ Use t-distribution (n < 30, σ unknown)

You have a sample of 100 catheters and are estimating the mean length using
sample standard deviation:
→ You could use Z (since n ≥ 30), but t is more precise

You are testing a drug and the population standard deviation is known (from past
trials):
→ Use Z-distribution
CQE Exam Tip:
Unless the population standard deviation is explicitly given, always assume you must use
the t-distribution—especially for small samples.
A political pollster wants to estimate voter preference with a margin of error of 5%. If p
̂ =
0.5 and α = 0.05, how many voters should they survey?
Great! This is a sample size calculation for estimating a population proportion, which is
common in polling and attribute data analysis — and very relevant for the ASQ CQE exam.
Given:
Parameter
Value
Estimated proportion (p^\hat{p}) 0.5
Margin of error (E)
5% = 0.05
Confidence level
95%
Significance level (α)
0.05
Z-value for 95% confidence
1.96
A market researcher is studying customer satisfaction. If they reduce their acceptable
margin of error from 5% to 2.5%, how will this a ect their required sample size?
Excellent question — this gets right to the heart of sample size sensitivity in statistical
planning, which is critical for both the ASQ CQE exam and real-world quality studies.
Conclusion:
Reducing the margin of error from 5% to 2.5% will increase the required sample size by a
factor of 4.
If the researcher originally needed 400 responses, they’ll now need 1,600.
CQE Tip:
This relationship helps you balance cost and precision — smaller margin = more precise =
more expensive (larger sample).
6B-2 Drawing Statistical Conclusions
6B-3 Probability Terms and Concepts
Classic Model:
Relative Frequency of Occurrence
Relative Frequency of Occurrence is one of the most practical ways to understand
probability, because it’s based on actual observations or experiments, not just theory.
Definition
Key Points
1. Empirical – It’s based on observed data, not assumptions.
2. Approximates Probability – As the number of trials increases, the relative frequency
approaches the true probability (by the Law of Large Numbers).
3. Used when theoretical probability is hard to calculate – For example, complex
systems where mathematical models are di icult.
Example
Imagine rolling a 6-sided die 100 times.
Outcome Times Occurred Relative Frequency
1
14
14/100 = 0.14
2
17
17/100 = 0.17
3
15
15/100 = 0.15
4
18
18/100 = 0.18
5
19
19/100 = 0.19
6
17
17/100 = 0.17
If you rolled it 1,000 times, the frequencies would get closer to the theoretical probability
(1/6 ≈ 0.1667 for each side).
Real-world Uses

Quality control: % of defective products in a batch

Weather forecasting: % of days it rained in the past 10 years in July

Sports analytics: Player’s batting average = hits / total at-bats
In short:
Relative frequency = observed probability from data.
The more trials you do, the closer it gets to the true probability.
Example: Rainy Days in a Month
Suppose you track the weather in your city for August (31 days) and count how many days it
rained.
Day Rain? (Yes/No)
1
Yes
2
No
3
No
4
Yes
…
…
31 Yes
At the end of the month, you find:

Number of days it rained: 12

Total days observed: 31
How this helps:
If you wanted to guess whether it will rain on a random day next August, you could use
0.387 as your probability estimate — even without knowing complex weather models.
Example: Factory Defects
A medical device factory produces orthopedic screws.
You inspect 500 screws from one week’s production and find 15 defective ones.
Step 1 – Count occurrences

Number of defective screws = 15

Total screws inspected = 500
Step 3 – Interpretation
This means:
“Based on this sample, about 3% of screws are defective.”
If you inspected more screws over time, this percentage might change slightly, but with
more data, it will stabilize near the true defect rate.
Step 4 – Application

Quality Control: Helps decide if production is within acceptable limits.

Predictive Use: If production next week is 10,000 screws, expected defective =
10,000×0.03=30010,000 \times 0.03 = 300 screws.
If you want, I can also show you how this links to the Law of Large Numbers with a short
simulation table — that’s a favorite in ASQ exam questions.



Experiment/Trial: Some thing done with an expectation of result.
Event or Outcome: Result of experiment
Sample Space: A sample space of an experiment is the set of all possible results of
that random experiment. {1, 2, 3, 4, 5, 6}
Union
Intersection
Mutually Exclusive Events
Independent Events
what will be the probability of getting tail if we toss two coins twice
Conditional Probability
Complementary Events
Normal Probability Distribution
1. What is Normal Probability Distribution?

The normal distribution is a continuous probability distribution that is symmetric
about the mean.

It is often called the bell curve because of its shape: high in the middle and tapering
o equally on both sides.

Many real-world data (like measurement errors, human height, blood pressure, test
scores) tend to follow this distribution.
2. Key Characteristics
1. Shape: Symmetrical, bell-shaped curve.
2. Mean = Median = Mode (all are at the center of the curve).
3. Tails: Extend indefinitely in both directions but never touch the horizontal axis.
4. Area under the curve = 1 (or 100%) → meaning it represents the total probability.
5. Spread determined by Standard Deviation (σ):
o
Small σ → narrow, tall curve (data tightly packed around mean).
o
Large σ → wide, flat curve (data more spread out).
3. Probability Density Function (PDF)
4. Standard Normal Distribution (Z-distribution)
5. The Empirical Rule (68-95-99.7 Rule)
For a normal distribution:

68% of data lies within ±1σ from the mean.

95% of data lies within ±2σ.

99.7% of data lies within ±3σ.
This is very useful in quality control (e.g., Six Sigma).
6. Example in Real Life
Suppose the tensile strength of a material follows a normal distribution with:

Mean (µ) = 50 MPa

Standard deviation (σ) = 5 MPa
Then:

About 68% of samples will have strength between 45 and 55 MPa.

About 95% between 40 and 60 MPa.

About 99.7% between 35 and 65 MPa.
7. Applications in Quality Engineering

Control charts: Assume process measurements follow normal distribution.

Process capability analysis (Cp, Cpk).

Tolerance intervals in design.

Hypothesis testing (Z-test, t-test).
In simple words:
The normal distribution is like a balance point of nature – most values cluster near the
mean, and extreme values become less and less likely. It helps us calculate probabilities,
detect variation, and set quality limits.
Standard Normal Distribution (Z-distribution),
This is where most of the probability calculations come from.
4. Standard Normal Distribution (Z-distribution)
What it means

The normal distribution can have any mean (µ) and any standard deviation (σ).

To make things simple and consistent, statisticians created a “standardized” version
of it:
o
Mean (µ) = 0
o
Standard deviation (σ) = 1
This is called the Standard Normal Distribution (or Z-distribution).
Why we need it

There are infinitely many possible normal curves (di erent µ and σ).

Instead of having separate probability tables for each, we convert every normal
distribution into the standard normal distribution.

Then, we can use Z-tables (pre-calculated areas under the curve) to find probabilities
easily.
How to Convert Any Normal Variable to Z (Standardization)
Example
Using Z-tables

Z-tables tell us the area under the curve to the left of a given Z.

For Z=1.6, the table gives 0.9452 → meaning 94.52% of values are below 58 MPa.

So, probability of strength greater than 58 MPa = 1−0.9452=0.05481 - 0.9452 = 0.0548
(about 5.5%).
Visualization
You can imagine this process as:
1. Take your data point.
2. Convert it to “how many σ away” (Z-score).
3. Look up that Z in the standard normal curve to find probability.
In simple terms:
The Z-distribution is just a “reference curve.”
We convert every normal problem into this standard curve so we can quickly calculate
probabilities without recalculating areas every time.
Probability between two values (45 and 55 MPa) in detail without the plot:
Problem Setup

Normal distribution: μ=50,σ=5μ = 50, σ = 5

Interval: 45 < X < 55
Step 1: Convert values to Z-scores
Step 2: Find probabilities from Z-table

P(Z<−1) = 0.1587

P(Z<1) = 0.8413
Step 3: Subtract to find the middle area
Final Answer: About 68.3% of rods will have tensile strength between 45 MPa and 55 MPa.
This exactly matches the Empirical Rule (68% within ±1σ).
Binomial Probability Distribution
1. What is it?

The binomial distribution describes the probability of getting a certain number of
“successes” in a fixed number of independent trials.

Each trial has only two possible outcomes:
o
Success (e.g., defective, heads, pass)
o
Failure (non-defective, tails, fail)
That’s why it’s called binomial (two outcomes).
2. Conditions for Binomial Distribution
It applies only if these are true:
1. Fixed number of trials (n)
Example: Testing 10 light bulbs.
2. Each trial has only 2 outcomes
Example: Defective or not defective.
3. Probability of success (p) is constant for every trial.
4. Trials are independent
Example: The outcome of one bulb test does not a ect another.
3. Probability Formula
4. Mean and Variance
5. Example
6. Shape of Distribution

If p=0.5, distribution is symmetric.

If p is very small or very large, distribution is skewed.

For large n, binomial distribution approximates the normal distribution (Central Limit
Theorem).
7. Applications in Quality Engineering

Defects in samples (e.g., probability of finding a certain number of defectives in
inspection).

Acceptance sampling plans.

Reliability testing (e.g., probability of a certain number of failures).

Yes/No surveys (success/failure outcomes).
In simple words:
The binomial distribution is about counting successes in a fixed number of trials when each
trial has only two outcomes and a constant probability.