Statistics Data (NJK)

Statistics Data

What Is Statistics?

Statistics is the branch of applied mathematics concerned with collecting, organising, presenting, analysing, and interpreting numerical facts so that sound engineering decisions can be made from them. On a shop floor, this is not academic — it's how a batch of turned shafts gets accepted or rejected, how a machine's wear pattern is tracked, and how a process is proven "in control."

Working definitionStatistics is the science of drawing reliable conclusions about a whole population (e.g. every shaft produced this month) from a manageable sample (e.g. 10 shafts measured on the lathe).

The five-stage process

Every statistical study — from a tolerance study to a failure analysis — moves through the same five stages.

01 Collection measure shafts 02 Organisation sort & tabulate 03 Presentation tables & charts 04 Analysis mean, median, SD 05 Interpretation accept / reject

Fig 1.1 — The five stages of a statistical study, from raw measurement to an engineering decision.

Why a mechanical engineer needs it

• Quality control: deciding whether a production batch meets tolerance using sample measurements.
• Process capability: quantifying how much a machine's output naturally varies.
• Reliability & failure analysis: predicting component life from test data (e.g. fatigue testing).
• Material testing: comparing average strength of different material batches.
• Experimentation: validating whether a design change actually improved performance, or whether the difference is just random scatter.

Data — Definition, Types & Uses

Data is the raw, unprocessed set of facts, figures, or measurements collected during an observation, test, or inspection — for example, the individual diameter readings taken off ten shafts with a micrometer, before anyone has averaged or sorted them.

Classification of data

DATA QUALITATIVE (attribute) e.g. surface finish: smooth / rough e.g. defect type: crack, burr, dent QUANTITATIVE (numerical) DISCRETE (countable) no. of defective bolts / batch CONTINUOUS (measured) shaft diameter (mm) By source, data is also grouped as: PRIMARY — collected first-hand by the engineer (own inspection log)   |   SECONDARY — obtained from existing records (supplier's material certificate)

Fig 2.1 — Data first splits by nature (qualitative vs quantitative), then quantitative data splits by countability; separately, all data is classed by source as primary or secondary.

Type	Definition	Mechanical engineering example
Qualitative	Describes a quality or attribute, not measured in numbers	Type of surface defect: pit, scratch, crack
Quantitative – Discrete	Countable, takes whole-number values only	Number of rejected components in a lot
Quantitative – Continuous	Measurable, can take any value within a range	Shaft diameter, tensile strength, temperature
Primary	Collected directly by the observer for the purpose at hand	Readings taken on-site with a micrometer
Secondary	Already collected by someone else, reused for analysis	Vendor's mill test certificate

Uses of statistics for a mechanical engineer

Quality control chartsTolerance & fit analysisProcess capability (Cp, Cpk)Material testingReliability & life predictionDesign of experimentsInventory & demand forecastingMaintenance scheduling

Measures of Central Tendency

A measure of central tendency is a single value that represents the centre, or typical value, of an entire data set — the number you'd quote if someone asked "so, roughly, what diameter are these shafts?"

Worked example — used throughout this sheetTen shafts turned on a lathe are measured with a micrometer. Diameter (mm): 25.1, 25.3, 25.2, 25.4, 25.2, 25.5, 25.2, 25.3, 25.6, 25.2

Mean (x̄)

The arithmetic average — sum of all observations divided by their count. It uses every value in the data set, which makes it sensitive to outliers (one badly oversized shaft pulls the mean up).

x̄ = ( Σx ) / nΣx = sum of all readings, n = number of readings

x̄ = 253.0 / 10 = 25.30 mm

Median

The middle value once data is arranged in ascending order. For an even count of readings, it's the average of the two middle values. It is not disturbed by an extreme outlier — useful when one reading is clearly a measurement error.

Sorted: 25.1, 25.2, 25.2, 25.2, 25.2, 25.3, 25.3, 25.4, 25.5, 25.6n = 10 (even) → Median = average of 5th & 6th values

Median = (25.2 + 25.3) / 2 = 25.25 mm

Mode

The value that occurs most frequently in the data set. For a production process, the mode often reflects the machine's "natural" setting.

25.2 mm occurs 4 times — more than any other reading → Mode = 25.20 mm

MODE 25.1 25.2 25.3 25.4 25.5 25.6 Shaft diameter (mm) 1 2 1 1 1 4 Mode

Fig 3.1 — Frequency of each measured diameter. The tallest bar (25.2 mm, orange outline) is the mode. Because a few oversized shafts pull the average up, Mean (25.30) > Median (25.25) > Mode (25.20) — the signature of a mildly right-skewed process.

Empirical relationship (moderately skewed data)Mode ≈ 3 × Median − 2 × Mean  →  3(25.25) − 2(25.30) = 25.15 mm — close to the actual mode of 25.20 mm, confirming the batch is only mildly skewed.

Measure	Best used when…	Weakness
Mean	Data is fairly symmetric, no wild outliers	Distorted by extreme values
Median	Data has outliers or is skewed	Ignores the exact magnitude of other values
Mode	You need the most "typical"/common setting	May not exist, or more than one may exist

Measures of Dispersion

Two batches of shafts can share the exact same mean diameter and still be very different in quality — one tightly clustered around the target, the other scattered wide. Dispersion measures this spread, and is exactly what a tolerance band or a process-capability index is built on.

LOW SPREAD (good process) x̄ HIGH SPREAD (worn machine / poor control) x̄

Fig 4.1 — Both rows have the same mean (x̄), but the bottom row is far more spread out. Central tendency alone would hide this difference — dispersion reveals it.

Range

The simplest measure — the difference between the largest and smallest observation.

Range = Maximum − Minimum = 25.6 − 25.1 = 0.5 mm

Deviation from the mean

Before computing variance, each reading's distance from the mean (25.30 mm) is found:

Reading (mm)	Deviation (x − x̄)	Squared deviation
25.1	−0.20	0.0400
25.2 (×4)	−0.10	0.0400 total
25.3 (×2)	0.00	0.0000
25.4	+0.10	0.0100
25.5	+0.20	0.0400
25.6	+0.30	0.0900
Σ(x − x̄)²		0.2200

Variance (σ²)

σ² = Σ(x − x̄)² / naverage of the squared deviations

σ² = 0.2200 / 10 = 0.0220 mm²

Standard Deviation (σ)

σ = √(σ²)back to the original units (mm)

σ = √0.0220 = 0.148 mm

Why standard deviation matters more than range on the shop floorRange only looks at the two extreme readings. Standard deviation accounts for every reading, which is why it's the basis for process-capability indices (Cp, Cpk) and control-chart limits (typically set at x̄ ± 3σ).

Measure	Formula	Result for our batch
Range	Max − Min	0.50 mm
Variance	Σ(x−x̄)² / n	0.0220 mm²
Standard Deviation	√Variance	0.148 mm