STAT1008 Chap.3 Measures of Centre, Variation and Shape
Measures of Centre, Variation and Shape
Measures of Centre, Variation and Shape turns data into numbers that can be compared.
It covers the mean, median and mode and when each is appropriate, the range, variance and standard deviation with the n − 1 divisor, z-scores and the plus or minus 3 outlier rule, the coefficient of skewness and its interpretation bands, the empirical rule for bell-shaped data, and covariance and correlation for two numerical variables. Every result is interpreted in the units of the data.
Questions typically ask for a hand calculation on a small dataset followed by an Excel check, then an interpretation that pairs a location measure with a spread measure. Shape decides which pair to quote, and the course’s skewness bands give a numerical way to justify that choice.
What this chapter covers
- 01
Mean, median and mode
- 02
Resistant and non-resistant measures
- 03
Variance and standard deviation
- 04
Z-scores and outliers
- 05
Skewness and the position of mean and median
- 06
The empirical rule and Chebyshev
- 07
Covariance and correlation
Worked example · free
Mean, median and standard deviation of weekly work hours
- 1Mean = 104/8 = 13 hours.
- 1Median = average of the 4th and 5th ordered values = (12 + 12)/2 = 12 hours.
- 2Squared deviations from 13 sum to 250, so s² = 250/7 = 35.71 hours².
- 1s = √35.71 = 5.98 hours.
- 1z = (26 − 13)/5.98 = 2.17, inside ±3, so not an outlier by the z-score rule.
Key terms
- Standard deviation
- The standard deviation, the positive square root of variance, measures typical distance from the mean in the data’s units.
- Z-score
- A z-score is the number of standard deviations an observation lies above or below the mean.
- Coefficient of skewness
- The coefficient of skewness is a number summarising asymmetry, positive for a long right tail and negative for a long left tail.
- Correlation coefficient
- The correlation coefficient is covariance divided by the product of the two standard deviations, always between −1 and 1.
- Resistant measure
- A resistant measure is a summary, such as the median or IQR, that changes little when an extreme value is added.
- Sample mean
- The sample mean adds the observations and divides by how many there are.
- Covariance
- Covariance is the average product of paired deviations from the two means, signalling the direction of a linear relationship.
Measures of Centre, Variation and Shape FAQ
Should I report the mean or the median?
Look at the shape first. For roughly symmetric data the mean and standard deviation summarise well. For skewed data or data with extreme values, the median and interquartile range are better, because a single large value can pull the mean a long way.
Why is the sample variance divided by n minus 1?
Dividing the sum of squared deviations by n minus 1 instead of n makes the sample variance an unbiased estimate of the population variance. Population formulas divide by N, and Excel separates them as VAR.S and VAR.P.
Does a correlation near zero mean the variables are unrelated?
Not necessarily. Correlation measures linear association only, so a strong curved relationship can still give a coefficient close to zero. Check the scatter plot, and remember that even a strong correlation does not establish cause.
What does a z-score of minus 2 mean?
It means the observation lies two standard deviations below the mean of its dataset. The sign gives the direction and the size gives the distance, so a z-score lets you compare values measured on different scales, such as two tests with different averages.
How is the empirical rule different from Chebyshev’s theorem?
The empirical rule applies only to bell-shaped data and gives approximate shares of 68, 95 and 99.7 per cent within one, two and three standard deviations. Chebyshev applies to any shape but only guarantees a lower bound, such as at least 75 per cent within two.
Exam move
Work one small dataset completely by hand each day: mean, median, mode, range, variance by both formulas, standard deviation, z-scores of the extremes and a sentence of interpretation. Then confirm every number in Excel. Learn the skewness bands and the mean, median and mode ordering for left and right skew as a picture you can sketch. Finish with one covariance-to-correlation conversion and a causation caveat.
Keep a sheet of the Excel functions beside the formulas they reproduce: AVERAGE, MEDIAN, MODE.SNGL, VAR.S, STDEV.S, SKEW, COVARIANCE.S and CORREL. Each time a hand answer disagrees with Excel, find the step that caused it before moving on, since that habit catches most errors before the exam does.
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