LGT5101 Chap.3 Centre, Spread, Shape and Standard Scores
Centre, Spread, Shape and Standard Scores
Summaries work as a set
Mean, median and mode describe different notions of centre. The mean uses every magnitude and therefore responds to a distant value. The median uses order and remains attached to the middle position. The mode identifies the most frequent value and can describe categorical data.
Choose the leading centre from the distribution shape and the decision: ordinary experience, total resource use and most common category are not the same question.
Pair every centre with spread
Range uses only the extremes, while variance and standard deviation use deviations from the mean. Standard deviation returns to the original units; variance uses squared units.
A narrow and a wide distribution can share the same mean, so a centre without spread gives a false sense of consistency. Report shape as well, because skewness and multiple clusters explain why a single numerical centre may be incomplete.
Keep sample and population scope visible
Sample formulas use sample notation and n minus one for variance; population parameters use Greek notation and N.
State which scope applies before using a calculator function. Preserve precision through squared deviations and round the reported result only at the end. Signed deviations from the mean should sum to zero apart from rounding, providing a quick arithmetic check.
Standardise without overclaiming
A z-score expresses one value as a signed number of standard deviations from its mean.
Sign gives direction and absolute size gives distance, but a z-score alone does not supply a tail probability without a distributional model. Empirical-rule percentages require an approximately bell-shaped distribution. Chebyshev supplies weaker guaranteed proportions for any shape. An extreme standard score is a signal to verify the record and process, not automatic permission to delete the observation.
Finish by explaining how the chosen centre, spread and shape affect the management decision.
What this chapter covers
- 01
Mean, median and mode
- 02
Outliers and robust centre
- 03
Range and standard deviation
- 04
Sample and population formulas
- 05
Skewness direction
- 06
Empirical rule
- 07
Chebyshev bounds
- 08
Z-scores
Worked example · free
Summarise one long packing job
- 3Calculate mean, median and mode.
- 4Calculate sample variance and standard deviation.
- 3Standardise and interpret the longest job.
Key terms
- Sample Mean
- The sum of sampled values divided by the sample size.
- Median
- The middle ordered value, or the average of the middle pair.
- Sample Variance
- Squared deviations from the sample mean divided by n minus one.
- Right Skew
- A distribution with a long high-value tail and mean above median.
- Z-score
- A value's signed distance from the mean in standard-deviation units.
Centre, Spread, Shape and Standard Scores FAQ
When should the median lead the report?
Lead with the median when a skewed distribution or distant observations make the mean unrepresentative of an ordinary case. Retain the mean when total resources or cumulative impact matter, because every magnitude still contributes to the total. The two centres answer different management questions.
Why does sample variance use n minus one?
The sample mean was estimated from the same observations, so the deviations have one fewer independent degree of freedom. Dividing by n minus one corrects the systematic downward bias that would arise if sampled squared deviations were divided by n.
When can I use the empirical rule?
Use its approximate 68%, 95% and 99.7% bands only for an approximately bell-shaped distribution. If the shape is unknown or clearly non-normal, Chebyshev provides a guaranteed minimum proportion within k standard deviations, although the bound is less precise.
Exam move
For three small datasets, calculate mean and median, sketch the shape and decide which centre answers an ordinary-case question. Recompute one dataset after adding a distant value and record how range and standard deviation move. Practise identifying whether a question describes a sample or population before writing the variance denominator, then use z-scores only after naming the relevant mean and standard deviation.
Create a calculation table with values, deviations and squared deviations; verify that signed deviations sum to zero and that squared deviations are non-negative. Report variance in squared units and standard deviation in original units. For each dataset, write separate sentences for ordinary experience and total resource use so the choice of centre follows the decision.
Next compare the empirical rule with Chebyshev at two standard deviations, stating the shape assumption that buys the sharper percentage. Treat any extreme z-score as an audit trigger: check transcription, measurement and process context before considering exclusion.
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