Quantitative: numbers — can average and find spread
3
Discrete: countable, exact values (whole numbers)
4
Continuous: measured, any value in a range
🧠
Key Words — What Do They Mean?
QualitativeA label or category
QuantitativeAn actual number
DiscreteCounted, whole steps
ContinuousMeasured, any value
🗂️
Quick Classify
Pick an example and click Classify.
📊 Statistics for T Level / Level 3 / Level 3 Engineering
Statistics is how engineers make sense of measurements, test results, and quality data. Every dimension you measure, every batch you test, every sensor reading — statistics turns numbers into decisions.
Mean, Median, ModeStandard DeviationHistograms & Box plotsCorrelationProbabilityQuality Control
🗂️
Types of Data
Knowing your data type determines which statistics to use, which charts to draw, and how to analyse results. Getting this wrong leads to wrong conclusions.
Qualitative (Categorical)
Describes categories or labels, not numbers you can do arithmetic with.
Nominal: no natural order → material type, colour, pass/fail
Ordinal: has natural order but gaps not equal → 1st/2nd/3rd, grade A/B/C
💡 Charts: bar chart, pie chart, tally chart. Cannot calculate a meaningful mean.
Quantitative (Numerical)
Actual numbers — you can add, subtract, find averages and standard deviation.
Discrete: only specific values (usually integers) → number of defects, number of parts
Continuous: any value in a range → diameter, temperature, voltage, mass
💡 Continuous data → histogram (bars touch). Discrete data → bar chart (bars separate).
📌 Engineering context: Most measurement data (dimensions, tolerances, forces, temperatures) is continuous. Defect counts and sample sizes are discrete. Choosing the right statistical method depends entirely on knowing which type you have.
📊
Measures of Central Tendency — Mean, Median, Mode
Three different ways to find the "typical" value of a dataset. Each tells a different story. In engineering, the mean is most common, but the median is more robust against outliers.
📋 Averages — Quick Reference
1
Mean = Σx ÷ n — add everything, divide by count
2
Median = order the data, pick the middle value
3
Mode = the value that appears most often
4
Outliers pull the mean — use median instead
🧠
Key Words — What Do They Mean?
MeanThe "average"
MedianThe middle value
ModeThe most common value
OutlierAn unusually high/low value
📊
Mean · Median · Mode Finder
Enter comma-separated numbers.
Mean = Σx/n Median = middle value (ordered) Mode = most frequent value
⚠️ A single outlier can pull the mean significantly. If a dataset has outliers (e.g. one very long repair time among normal ones), the median is more representative than the mean.
📐
Measures of Spread — Range, IQR, Variance, Standard Deviation
Two datasets can have the same mean but very different spreads. The standard deviation tells you how tightly clustered data is around the mean — essential for quality control.
📋 Spread — Quick Reference
1
Range = max − min (simplest, sensitive to outliers)
2
IQR = Q3 − Q1 (middle 50%, robust to outliers)
3
Variance s² = Σ(x−x̄)² ÷ (n−1)
4
Standard deviation s = √(variance)
🧠
Key Words — What Do They Mean?
RangeBiggest minus smallest
QuartileOne of 4 equal data chunks
VarianceAverage squared distance from mean
Std. DeviationHow spread out the data is
📐
Range & SD Finder
Enter comma-separated numbers.
σ = √[ Σ(x − x̄)² / n ] (population) s = √[ Σ(x − x̄)² / (n−1) ] (sample, use this one)
Standard Deviation — step by step
Step 1: Find the mean x̄ = Σx/n
Step 2: Find each (x − x̄)²
Step 3: Sum them: Σ(x − x̄)²
Step 4: Divide by (n−1): variance s²
Step 5: Take √: standard deviation s
Data: 12, 14, 15, 16, 13 (n=5)
Mean: x̄=(12+14+15+16+13)/5=70/5=14
(x−x̄)²: 4, 0, 1, 4, 1
Σ(x−x̄)²: 10
s²=10/(5−1): 2.5
s=√2.5:1.58 ✅
Range and IQR
Range = Max − Min (simplest, affected by outliers)
IQR = Q3 − Q1 (middle 50%, robust to outliers)
Q1 = lower quartile (25th percentile)
Q3 = upper quartile (75th percentile)
Ordered data: 10,12,14,15,16,18,20
Range: 20−10=10
Q1: median of {10,12,14}=12
Q3: median of {16,18,20}=18
IQR: 18−12=6 ✅
💡 An outlier is typically defined as any value below Q1−1.5×IQR or above Q3+1.5×IQR.
📋
Grouped Data — Frequency Tables
When data is too numerous to list individually, it is grouped into class intervals. You estimate the mean and standard deviation using midpoints of each class.
50 bolts were measured. Find the estimated mean and standard deviation.
Class (mm)
Mid (m)
Freq (f)
f × m
(m − x̄)²
f(m − x̄)²
10 – 12
11
3
33
(11−16.32)²=28.27
84.82
12 – 14
13
7
91
(13−16.32)²=11.02
77.17
14 – 16
15
12
180
(15−16.32)²=1.74
20.90
16 – 18
17
15
255
(17−16.32)²=0.46
6.96
18 – 20
19
8
152
(19−16.32)²=7.18
57.47
20 – 22
21
5
105
(21−16.32)²=21.90
109.49
Total
—
50
816
—
356.81
Estimated mean
x̄ = Σ(f×m) / Σf = 816/50
Answer:x̄ = 16.32 mm ✅
Estimated standard deviation
s² = 356.81 / (50−1) = 7.28
s =√7.28 = 2.70 mm ✅
📌 These are estimates because we assumed all values in a class lie at the midpoint. The true values would give a slightly different answer.
📈
Charts and Graphs
Visual representations of data. Choosing the right chart for the data type is as important as the calculation itself.
📋 Charts — Quick Reference
1
Frequency density = frequency ÷ class width
2
Histogram bars touch — continuous data, no gaps
3
Area of a bar = frequency (not the height!)
🧠
Key Words — What Do They Mean?
HistogramBars touching, continuous data
Freq. DensityFreq ÷ class width
Modal ClassClass with the tallest bar
📈
Frequency Density Finder
FrequencyClass width
Enter frequency & class width.
Histogram rules
Bars TOUCH (continuous data — no gaps)
y-axis = Frequency Density = freq ÷ class width
Area of bar = frequency (not height!)
Modal class = class with greatest frequency density
⚠️ Area = frequency means bars with wider classes are shorter than their frequency suggests. Always use freq density, not raw frequency, for the y-axis.
Choosing the right chart
Chart type
Use for
Bar chart
Discrete / categorical data
Histogram
Continuous grouped data
Pie chart
Proportions / categorical
Scatter graph
Relationship between two variables
Line graph
Continuous change over time
Box plot
Distribution, quartiles, outliers
〜
Cumulative Frequency and Box Plots
Cumulative frequency lets you read off medians, quartiles and percentiles directly from a graph. The box plot summarises the entire distribution in one compact diagram.
📋 Cumulative Frequency — Quick Reference
1
Median read at CF = n ÷ 2
2
Q1 read at CF = n ÷ 4
3
Q3 read at CF = 3n ÷ 4
4
Box plot box = Q1 to Q3, whiskers = min/max
🧠
Key Words — What Do They Mean?
CumulativeRunning total
Box PlotPicture showing quartiles
WhiskerLine to min or max
〜
Quartile Position Finder
n (total data points)
Enter n to find CF read-off points.
How to read a CF curve
1️⃣ Read up from x-axis to the curve
2️⃣ Read across to y-axis for cumulative frequency
3️⃣ Median: read at CF = n/2
4️⃣ Q1: read at CF = n/4
5️⃣ Q3: read at CF = 3n/4
💡 The CF curve is always an S-shape (sigmoidal). It starts at (min, 0) and ends at (max, n), and is always increasing.
Box plot interpretation
Box spans Q1 to Q3 = middle 50% of data
Median line inside box (Q2)
Whiskers extend to min and max
IQR = Q3 − Q1 = width of box
Skewness: if median is left of centre → right-skewed
📌 Outliers are sometimes shown as individual crosses (×) beyond the whiskers, defined as more than 1.5×IQR from Q1 or Q3.
⋯
Scatter Graphs and Correlation
A scatter graph shows the relationship between two variables. The Pearson correlation coefficient r measures the strength and direction of a linear relationship. The line of best fit lets you predict one variable from the other.
📋 Scatter & Correlation — Quick Reference
1
r ranges from −1 (perfect negative) to +1 (perfect positive)
2
Line of best fit: y = mx + c
3
Correlation ≠ causation!
4
Interpolation (inside range) is reliable; extrapolation is not
🧠
Key Words — What Do They Mean?
CorrelationHow two things move together
Best Fit LineStraight line through the points
InterpolatePredict inside known range
ExtrapolatePredict outside known range
⋯
Line of Best Fit Predictor
mcx
y = mx + c — enter values.
r = Σ(x−x̄)(y−ȳ) / √[Σ(x−x̄)²·Σ(y−ȳ)²] r∈[−1,+1] Line of best fit: y = mx + c
Correlation strength guide
r value
Strength
Direction
r = +1
Perfect
Positive
0.7 to 1
Strong
Positive
0.3 to 0.7
Moderate
Positive
0 to 0.3
Weak
Positive
r = 0
No correlation
—
−0.3 to 0
Weak
Negative
r = −1
Perfect
Negative
⚠️ Correlation does NOT mean causation! Two things can move together by coincidence or because of a third hidden factor.
Line of best fit
y = mx + c (equation of the line)
m = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)²
c = ȳ − m·x̄
Line always passes through (x̄, ȳ)
From scatter data: x̄=50, ȳ=122.7, m=0.750
Find c: c=122.7−0.750×50=122.7−37.5=85.2
Predict R at T=55°C: R=0.750×55+85.2=126.5Ω ✅
💡 Interpolation: predicting within the data range — reliable. Extrapolation: predicting outside — unreliable, use with caution!
🎲
Probability
Probability quantifies likelihood — from 0 (impossible) to 1 (certain). In engineering: quality acceptance sampling, fault rates, reliability analysis.
📋 Probability — Quick Reference
1
AND rule (independent): P(A and B) = P(A) × P(B)
2
OR rule (mutually exclusive): P(A or B) = P(A) + P(B)
3
Complement: P(not A) = 1 − P(A)
🧠
Key Words — What Do They Mean?
ProbabilityHow likely something is
IndependentOne doesn't affect the other
ComplementThe event NOT happening
🎲
Probability Combiner
P(A)P(B)
Enter two probabilities (0–1).
P(event) = favourable outcomes / total outcomes 0 ≤ P(A) ≤ 1 P(not A) = 1 − P(A)
Combining probabilities
AND rule (independent): P(A and B) = P(A) × P(B)
OR rule (mutually exclusive): P(A or B) = P(A) + P(B)
OR rule (general): P(A or B) = P(A)+P(B)−P(A and B)
Complement: P(not A) = 1 − P(A)
Component fails with P=0.05. Two independent. Find P(both fail).
Statistical thinking underpins every aspect of engineering quality, safety, and reliability. These are the most tested contexts in T Level / Level 3 assessments.
📋 Engineering Statistics — Quick Reference
1
Cp = (USL − LSL) ÷ 6σ (process capability)
2
Control limits = x̄ ± 3σ/√n
3
MTBF = total hours ÷ number of failures
🧠
Key Words — What Do They Mean?
Cp / CpkCan the process hit the tolerance?
MTBFAverage time between failures
Control ChartGraph that flags process problems
🔩
Process Capability (Cp) Calculator
USLLSLσ
Cp = (USL−LSL)/(6σ). Cp>1.33 is generally acceptable.
⚙️ Statistical Process Control (SPC)
Manufacturing processes produce variation. SPC uses control charts to distinguish normal variation from special cause variation that needs fixing.
Control limits (for means chart): UCL = x̄ + 3σ/√n LCL = x̄ − 3σ/√n
Instead of inspecting every part, a sample is tested. If the sample passes, the batch is accepted. Statistics tells you the risk of letting bad batches through.
A grouped frequency table has classes 20-30 (f=5), 30-40 (f=8), 40-50 (f=12), 50-60 (f=6), 60-70 (f=4). Find the estimated mean.
[4]
Midpoints: 25,35,45,55,65 Σ(f×mid)=5×25+8×35+12×45+6×55+4×65=125+280+540+330+260=1535 Σf=35 Mean = 1535/35 = 43.9 ✅
Q4
State whether each dataset is discrete or continuous: (a) number of faulty components in a batch, (b) diameter of a shaft in mm, (c) number of employees absent, (d) voltage reading in volts.
[4]
(a) Discrete — whole numbers only (b) Continuous — any value (c) Discrete — whole numbers (d) Continuous — any value
🟡 Tier 2 — Applied Problems
Q5
A scatter graph shows: x̄=40, ȳ=85, Σ(x−x̄)(y−ȳ)=360, Σ(x−x̄)²=120. Find m and c for the line of best fit.
[4]
m = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² = 360/120 = 3 c = ȳ−m·x̄ = 85−3×40 = 85−120 = −35 Line: y=3x−35 ✅
Q6
A batch of 200 components has 4% defective. A sample of 10 is taken. Find (a) the expected number of defects, (b) P(zero defects in sample).
From a cumulative frequency curve for 80 data points: Q1=24, Q3=36. (a) Find IQR, (b) identify if 55 is an outlier (using 1.5×IQR rule), (c) find the median from CF=40.
[5]
(a) IQR = 36−24 = 12 (b) Upper fence = 36+1.5×12 = 54. Since 55>54, 55 IS an outlier ✅ (c) CF=40 = n/2, so median is read from graph at CF=40 (value depends on graph)
🔴 Tier 3 — Engineering Challenge
Q8
A process produces shafts with mean 50mm, SD 0.8mm. The specification is 50±2mm. (a) Calculate Cp. (b) Find the probability that a single shaft is outside specification, using the rule that 99.7% of data lies within ±3σ of the mean.
[5]
(a) Cp=(USL−LSL)/(6σ)=(52−48)/(6×0.8)=4/4.8=0.83 (below 1.0, not capable ✗) (b) ±3σ = ±2.4mm, spec is ±2mm which is only ±2.5σ Within ±2.5σ: ≈98.76% → outside: ≈1.24%
Q9
Tensile strength data (MPa): 410, 425, 418, 432, 415, 420. Find mean, standard deviation, and state the 95% confidence interval (use x̄ ± 2s/√n).