ETF1100 Chap.4 Numerical Summaries and Distributions
Numerical Summaries and Distributions
Numerical Summaries and Distributions treats centre, spread and shape as one description. The chapter compares mean with median and standard deviation with interquartile range, shows how an outlier affects non-resistant statistics, introduces z-scores and the normal model, and insists that summary choice follow the observed distribution and business purpose rather than a memorised default.
What this chapter covers
- 01
Mean and median
- 02
Range and interquartile range
- 03
Variance and standard deviation
- 04
Skewness and outliers
- 05
Z-scores and standardisation
- 06
Normal distribution and model checks
Worked example · free
Measure the effect of an extreme value
- 1Add the values to obtain a total of 98 minutes.
- 1Compute mean = 98 / 7 = 14.0 minutes.
- 1Identify the fourth ordered value, giving median = 10 minutes.
- 1Describe the distribution as right-skewed because of the 38-minute case.
- 1Report the median as typical while investigating the valid operational exception.
Key terms
- Mean
- The arithmetic average obtained by dividing the total by the number of observations.
- Median
- The middle value of ordered data, or the average of the two middle values.
- Interquartile Range
- The upper quartile minus the lower quartile, measuring the middle half of a distribution.
- Standard Deviation
- A measure of typical distance from the mean in the original unit.
- Z-score
- An observation's signed distance from the mean measured in standard deviations.
- Outlier
- An observation unusually distant from the main pattern and requiring investigation.
Numerical Summaries and Distributions FAQ
When should I use the median instead of the mean?
Use the median as the typical centre when the distribution is strongly skewed or contains influential outliers. Pair it with the interquartile range. The mean may still matter for totals or resource planning, so explain the business purpose instead of applying a mechanical rule.
What does a z-score tell me?
It gives an observation's direction and distance from the mean in standard-deviation units. A z-score of 2 is two standard deviations above the mean. It is unit-free, but unusualness still depends on the distribution and whether a normal model is suitable.
Should every outlier be removed?
No. Check the source record and business context first. Correct a verified data error, retain and explain a valid unusual case, or analyse a separate process when appropriate. Removing a value merely because it changes the result biases the analysis.
Which units belong on the main summary measures?
Mean, median, range, interquartile range and standard deviation use the original measurement unit. Variance uses squared units, while a z-score is unitless. Tracking units is a fast reasonableness check: a numerical answer with the wrong dimensional form cannot correctly describe the quantity requested.
Exam move
Use a distribution-first routine. Sketch or inspect the histogram, describe symmetry or skewness, identify gaps, clusters and unusual values, then choose summaries. For symmetric data without strong outliers, practise mean with standard deviation; for skewed data, practise median with interquartile range. Keep units on every summary except z-scores, and remember that variance uses squared units.
Recompute one dataset before and after an extreme value so you can explain resistance rather than just name it. For normal-model questions, write z = observed minus mean divided by standard deviation, retain the sign and interpret distance before seeking an area. Do not assume that standardising makes a distribution normal. In business answers, report both the typical pattern and the operational meaning of valid extremes.
Keep full precision until final rounding, and state whether a result describes individual observations or the distribution as a whole. Create three small datasets with the same mean but different spread and shape, then sketch their ordered values or histograms. This separates the idea of centre from variability more effectively than memorising definitions.
For each dataset calculate the five-number summary, range, interquartile range, mean and standard deviation, and predict which measures will move if the largest value doubles. Check an apparent outlier against the original record and process notes before deciding whether it is an error, a valid exception or evidence of a second operating regime. If it remains valid, report it rather than trimming it for convenience.
Practise comparing groups only after confirming common units, time windows and inclusion rules. When using a normal model, mark the mean, standard-deviation steps and target value on a number line; the sign of the z-score should then be obvious. Translate the standardised result back to the original unit before making a business recommendation.
Ask whether a tail probability concerns individual outcomes or an assumed distribution, and do not read normal percentages from a visibly skewed sample without justification. On automatic questions, estimate the direction and rough magnitude first so a sign error or impossible negative spread is caught before submission. Finish with a sentence connecting shape, centre, spread and an operational consequence.
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