Statistics · Lesson 6 of 6
Chapter Summary and Practice
“Connect every measure of dispersion, select a calculation method, and reason through mixed data problems.”
• Connect measures of centre with the need to describe dispersion. • Compare range, mean deviation, variance and standard deviation, including their units and limitations. • Choose an appropriate method for raw, discrete-frequency and continuous grouped data. • Explain assumptions, coding transformations and the effects of shifting and scaling. • Solve mixed problems involving missing observations, replacement errors and omissions.
Read a dataset through both centre and spread
The chapter began with two players whose means and medians were equal but whose performances were distributed very differently. That example captures the central lesson: a representative value does not describe the whole dataset. A good interpretation connects a measure of centre with a suitable measure of dispersion and with the way the observations were collected or grouped.
Range asks how far apart the extremes are. Mean deviation averages distances from a stated centre. Variance averages squared deviations from the mean, and standard deviation takes its nonnegative square root. Each compresses information in a different way. A result should therefore name the statistic, the centre where needed, and the unit, rather than report an unexplained number.
| Measure | Meaning | Calculation | Unit | Limitation or caution |
|---|---|---|---|---|
| Range | Span from smallest to largest | Maximum − minimum | Original unit | Uses only two extremes |
| Mean deviation about the mean | Average absolute distance from x̄ | Σ|xᵢ − x̄|/n, or Σfᵢ|xᵢ − x̄|/N | Original unit | Use the actual mean; apply absolute value to each deviation |
| Mean deviation about the median | Average absolute distance from M | Σ|xᵢ − M|/n, or Σfᵢ|xᵢ − M|/N | Original unit | Find the median correctly; grouped medians require interpolation |
| Variance | Average squared distance from x̄ | Σ(xᵢ − x̄)²/n, or Σfᵢ(xᵢ − x̄)²/N | Squared original unit | Larger deviations receive greater weight; use the correct observation count |
| Standard deviation | Square root of average squared distance from x̄ | Nonnegative square root of variance | Original unit | Distinct from mean deviation; does not reconstruct the full distribution |
The chapter names quartile deviation but does not develop it. Our comparison focuses on the measures actually taught. All these measures describe numerical data in context; none alone tells us why a dataset is spread out or whether one process is preferable to another.
A coverage guide for the complete chapter
The main ideas form a connected progression. Cancellation explains why signs must be handled; frequencies explain weighting; grouping explains approximation; coding explains simpler arithmetic; and totals explain corrections. Use this guide to revisit the reasoning that supports the formulas rather than treating the chapter as a list to memorise.
| Chapter idea | What to understand | A useful self-check |
|---|---|---|
| Same centre, different scatter | Mean and median cannot determine spread | Can I construct equal-mean lists with different ranges? |
| Signed and absolute deviations | Signed deviations from the mean total zero | Did I use Σ|deviation| rather than |Σdeviation|? |
| Frequency weighting | Each row represents repeated observations | Did I divide by total frequency N? |
| Discrete medians | Median depends on positions in the expanded ordered data | For even N, did I locate both middle observations? |
| Continuous classes | Midpoints and interpolation approximate hidden raw values | Are boundaries continuous and is the median-class width correct? |
| Variance and standard deviation | Average squares, then take a nonnegative square root | Are the units squared for variance and original for standard deviation? |
| Second-moment identity | Expand centred squares and use Σfᵢxᵢ = Nx̄ | Have I distinguished Σfᵢxᵢ² from (Σfᵢxᵢ)²? |
| Step deviations | Subtract an origin and divide by a scale, then restore units | Did variance get h² and standard deviation h? |
| Transformations | Uniform shifts cancel from deviations; scales do not | Did I use a² for variance and |a| for standard deviation? |
| Missing and incorrect entries | Mean and variance encode a sum and squared sum | Did replacement keep n fixed and omission reduce it? |
The formulas share the same building blocks
Use n for the number of raw observations, fᵢ for frequencies and N = Σfᵢ for their total. In a discrete table xᵢ is an observed value; in a continuous grouped table it is a midpoint. The same frequency formulas apply to both representations, but only the discrete values retain the exact observations.
A consistent procedure reduces errors. Identify the data representation, determine the required centre or totals, select a convenient calculation, and check the result against its meaning. If class intervals are supplied, acknowledge the midpoint assumption. If statistics are supplied without data, recover S and Q rather than inventing observations.
| Information supplied | Useful method | Final checks |
|---|---|---|
| A short raw list | Mean or ordered median; a deviation table | Every repeated observation is included |
| Distinct values and frequencies | Weighted mean and deviation columns | N = total frequency, not row count |
| Intervals and frequencies | Boundaries, midpoints and weighted calculations | Grouping assumptions and original units stated |
| A fractional mean with exact value totals | Second-moment variance identity | Keep the mean exact until final subtraction |
| Large regularly spaced values | Assumed mean and step-deviation coding | Restore h for σ or h² for σ² |
| A uniform transformation | Shift and scale rules | Negative multipliers require an absolute value for σ |
| Mean, variance and some missing entries | Recover S and Q, then subtract known contributions | All proposed missing values satisfy both totals |
| Mistaken entries | Correct S, Q and possibly count | Use the corrected mean in the variance calculation |
A single raw dataset connects all four measures
A small list can show how the same observations yield different summaries. Calculate the centre before applying deviation formulas, and distinguish an absolute-deviation total from a squared-deviation total. Compare the final numbers by their definitions rather than expecting them to match.
Problem
For 1, 2, 2, 3, 12, find the mean, median and the principal measures of dispersion.
- 1.The total is 20 across five observations, so x̄ = 4. The ordered middle observation is M = 2. Range is 12 − 1 = 11.
- 2.Distances from the mean 4 are 3, 2, 2, 1, 8, totalling 16. Hence MD(x̄) = 16/5 = 3.2.
- 3.Distances from the median 2 are 1, 0, 0, 1, 10, totalling 12. Hence MD(M) = 12/5 = 2.4.
- 4.Squared deviations from the mean are 9, 4, 4, 1, 64, totalling 82. Variance is 82/5 = 16.4.
- 5.Standard deviation is √16.4 ≈ 4.05. Its unit matches the data, while variance has the squared unit.
- 6.The high observation 12 contributes strongly to both distance totals, especially the squared one. The lower median-based mean deviation does not mean the dataset itself changed.
| Value | |xᵢ − 4| | |xᵢ − 2| | (xᵢ − 4)² |
|---|---|---|---|
| 1 | 3 | 1 | 9 |
| 2 | 2 | 0 | 4 |
| 2 | 2 | 0 | 4 |
| 3 | 1 | 1 | 1 |
| 12 | 8 | 10 | 64 |
| Total | 16 | 12 | 82 |
Frequency tables preserve counts and positions
A compressed table must retain the same centre and spread as the expanded observations. Weighted totals handle repetition, while cumulative counts locate the median. This mixed example uses a fractional mean and a simpler median to show why each distance calculation must use the requested centre.
Problem
Values 1, 3 and 5 have frequencies 2, 3 and 1. Find the mean, median, both mean deviations and variance.
- 1.N = 6 and S = 2 × 1 + 3 × 3 + 1 × 5 = 16, so x̄ = 8/3.
- 2.Expanded ordered data are 1, 1, 3, 3, 3, 5. The third and fourth observations are both 3, giving M = 3.
- 3.Weighted absolute distances from 8/3 total 2 × 5/3 + 3 × 1/3 + 1 × 7/3 = 20/3. Divide by 6 to obtain MD(x̄) = 10/9.
- 4.Distances from 3 total 2 × 2 + 3 × 0 + 1 × 2 = 6, giving MD(M) = 1.
- 5.Squared-value total Q = 2 × 1² + 3 × 3² + 1 × 5² = 54. Variance = 54/6 − (8/3)² = 9 − 64/9 = 17/9.
- 6.Standard deviation is √17/3 ≈ 1.37. Every denominator comes from the six observations, not the three table rows.
Grouped results depend on their assumptions
Midpoint calculations and grouped median interpolation use different assumptions about the unseen positions within classes. Midpoints place each class’s observations at its centre for calculation. Median interpolation spreads the count evenly through the median class. State these assumptions when interpreting a grouped result, and do not claim the original raw observations must have had the same statistics.
Problem
Classes 0–10, 10–20, 20–30 have frequencies 2, 4, 2. Find grouped mean, median, mean deviation about the mean, variance and standard deviation.
- 1.Midpoints are 5, 15 and 25 with N = 8. Weighted total is 10 + 60 + 50 = 120, so the grouped mean is 15.
- 2.Cumulative frequencies are 2, 6, 8. The median class is 10–20, giving M = 10 + (4 − 2) × 10/4 = 15.
- 3.Weighted distances from 15 total 2 × 10 + 4 × 0 + 2 × 10 = 40, so MD(x̄) = 5.
- 4.Weighted squared deviations total 2 × 100 + 4 × 0 + 2 × 100 = 400. Variance is 400/8 = 50.
- 5.Standard deviation is √50 = 5√2 ≈ 7.07. Equality of the mean and estimated median here comes from this symmetric distribution, not a rule for every grouped table.
Coding is a change of coordinates for the arithmetic
A shortcut should be traceable back to the original observations. Subtracting an assumed mean moves the origin; dividing by h changes the scale. The coded mean needs both operations reversed, whereas centred spread is unaffected by the origin and only needs its scale restored.
Problem
Find the mean, variance and standard deviation of 100, 102, 104, 106, 108 by coding.
- 1.Choose A = 104 and h = 2. Codes are −2, −1, 0, 1, 2.
- 2.Their mean is zero, and their average square is (4 + 1 + 0 + 1 + 4)/5 = 2. Thus coded variance is 2.
- 3.Original mean is 104 + 2 × 0 = 104. Original variance is 2² × 2 = 8.
- 4.Standard deviation is √8 = 2√2 ≈ 2.83. Adding 104 to a coded standard deviation would be an incorrect reversal of the coding.
Uniform changes and isolated corrections are different operations
A transformation changes every observation according to one rule, so it changes all deviations predictably. A correction changes only specified entries; it requires updated totals and often an updated mean. Keep these two kinds of task distinct before selecting a formula.
Problem
A dataset has mean 50 and standard deviation 4. Every value is changed by yᵢ = −2xᵢ + 10. Find the new mean and spread.
- 1.The new mean is −2 × 50 + 10 = −90.
- 2.Original variance is 4² = 16. The new variance is (−2)² × 16 = 64.
- 3.New standard deviation is |−2| × 4 = 8.
- 4.The shift 10 affects the mean but not the centred spread. The negative scale reverses order without making standard deviation negative.
Problem
Ten observations have mean 5 and standard deviation 2. A recorded value 7 is corrected to 3, or alternatively removed. Find the new statistics in each case.
- 1.Recover original totals: S = 50 and Q = 10(4 + 25) = 290.
- 2.Replacement by 3 gives n = 10, S′ = 50 − 7 + 3 = 46 and Q′ = 290 − 49 + 9 = 250.
- 3.The replacement mean is 4.6; variance is 250/10 − 4.6² = 3.84 and standard deviation is √3.84 ≈ 1.96.
- 4.For omission, n′ = 9, S′ = 43 and Q′ = 241.
- 5.The omission mean is 43/9; variance is 241/9 − (43/9)² = 320/81, and standard deviation is √320/9 ≈ 1.99.
- 6.Both calculations remove the old recorded value and its square, but only the replacement adds a new value and retains the original count.
Checks that reveal a mistaken calculation
Each statistic has mathematical constraints and a clear meaning. Range and the dispersion measures are nonnegative. A signed-deviation total about the exact mean is zero. The cumulative-frequency total must equal N. Mean and median must fit within the observed extremes, or within appropriate boundaries for a grouped estimate.
Units provide another check: mean deviation and standard deviation use the original unit, and variance uses its square. Under a uniform shift only the mean moves; under a uniform scale the standard deviation changes by the multiplier’s magnitude. For corrections, track count, S and Q separately. A plausible-looking answer can still be wrong if any one of these relationships is violated.
Do not compare centre alone, divide by the number of table rows, select only one middle value for every even N, treat all grouped values as exact, confuse the square of a sum with the sum of squares, round the mean too early, forget a coding scale factor, or leave the old count after an omission.
Statistics grew from records into a mathematical discipline
The chapter’s historical note links the name of statistics with vocabulary about the state, including the Latin word status. It discusses early Egyptian census traditions and connects the subject with administrative records and information about births and deaths. Indian traditions of state data collection are discussed through the Arthashastra and records of surveys during Akbar’s reign. These examples explain why organising and interpreting data became useful long before the modern formulas were standard.
John Graunt studied birth and death records and helped develop vital statistics. Jacob Bernoulli’s work connected statistical reasoning with probability. Later, Francis Galton, Karl Pearson and Ronald Fisher contributed to statistical methods and their use in fields such as biology, agriculture and education. Their wider work provides historical context; the present chapter’s task is to understand and calculate dispersion.
Check your understanding
This final set spans the whole chapter. Use each answer to check a concept or a decision: the right centre, representation, denominator, formula, scale factor, or correction procedure matters as much as the arithmetic.
Quiz
Which measure uses only the two observed extremes?
For 1, 2, 2, 3, 12, what is the median?
For the same data, which is MD about the mean 4?
Values 1, 3, 5 have frequencies 2, 3, 1. What is N?
Which unit belongs to variance of heights measured in cm?
Codes have variance 2 and h = 2. What is variance in the original scale?
A standard deviation of 4 becomes what after multiplying every value by −2 and adding 10?
Ten values have mean 5 and standard deviation 2. What is their sum of squares?
A single incorrect entry is replaced by its correct value. What happens to count?
Why is a midpoint-based grouped variance an estimate of raw-data variance?
Practice Problems
- Create two equal-mean datasets with different ranges and compare their mean deviations and standard deviations.
- For 2, 3, 3, 7, 10, find the mean, median, range, mean deviation about each centre, variance and standard deviation.
- Values 2, 5, 8 have frequencies 3, 4, 3. Find weighted mean, median, both mean deviations and variance.
- Classes 10–20, 20–30, 30–40, 40–50 have frequencies 3, 7, 6, 4. Estimate the grouped mean and median, then calculate mean deviation about each and standard deviation.
- Inclusive classes 11–15, 16–20, 21–25, 26–30 have frequencies 4, 8, 6, 2. Convert boundaries and use coding to find the mean and standard deviation.
- Derive the second-moment variance identity and show how it leads to S = nx̄ and Q = n(σ² + x̄²).
- For 200, 204, 208, 212, 216, choose a convenient assumed mean and scale, then find mean, variance and standard deviation.
- A dataset has mean 12 and standard deviation 5. Find its mean and spread after adding −7 to every value, and after multiplying every value by −3.
- Five observations have mean 6 and variance 10. Three are 2, 5 and 7. Find the remaining two using both totals.
- Twenty observations have mean 10 and standard deviation 2. An entry 8 is omitted, or replaced by 12. Set up both corrected calculations and explain the different denominators.
- Explain the midpoint and interpolation assumptions in grouped data, and name an interpretation each assumption prevents you from claiming.
- Diagnose these errors: averaging signed deviations as spread; using the number of classes as N; taking a negative standard deviation; restoring a coded variance with h instead of h².
Key Takeaways
• A complete data interpretation connects centre, dispersion, representation and context. • Range, mean deviation, variance and standard deviation answer related but distinct questions about spread. • Frequency calculations preserve repetition; continuous grouped calculations estimate statistics through midpoints and interpolation. • Derivations explain cancellation, averaging, square roots, second-moment identities and coding scale factors. • Uniform shifts and scales have predictable effects; isolated corrections require updated count, sum and squared sum. • Check nonnegativity, units, counts, assumptions and precision before accepting a result.
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How Data Changes: Scaling and Corrections
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