Adam Mitchell, Human Resources Manager of Sydney First Bank (FB), is reviewing the employee training programs of SFBs. His staff tabulated the following statistics on the teller’s training hours. [table]
| Northern Suburbs | Southern Suburbs Mean | 35 | 30 Median | 30 | 30 Mode | 28 | 30 Standard deviation | 10 | 8
[/table] Which of the following statement is correct?

Question

BlackTom AI · Accepted Answer

When mean = median = mode, the distribution is (approximately) symmetric, so this key test immediately identifies the Southern Suburbs. 1) Relevant knowledge points and concepts - Measures of central tendency: mean, median, mode. - Relation to skewness: in a symmetric, unimodal distribution mean = median = mode. - Skew direction: if mean > median > mode, distribution is right (positive) skewed; if mean < median < mode, left (negative) skewed. - Dispersion: compare standard deviations to judge spread. 2) Step-by-step reasoning - Southern Suburbs: mean = 30, median = 30, mode = 30. All three central measures are equal, which is the textbook signature of a symmetric (unskewed) unimodal distribution. The standard deviation (8) is just a measure of spread but does not contradict symmetry. - Northern Suburbs: mean = 35, median = 30, mode = 28. The ordering is mean (35) > median (30) > mode (28), which matches the pattern of a right-skewed (positively skewed) distribution, not left-skewed. - Compare dispersions: Northern SD = 10, Southern SD = 8, so Northern is more dispersed (10 > 8). 3) Calculations / comparisons shown - Southern: mean = median = mode = 30 → equality implies symmetry. - Northern: 35 (mean) > 30 (median) > 28 (mode) → pattern indicates right skew. - Dispersion: 10 (Northern SD) vs 8 (Southern SD) → Northern has greater spread. 4) Why the correct answer is correct - Because mean, median and mode are all equal (30) for the Southern Suburbs, the distribution is symmetric. This is the defining relationship used to infer symmetry from summary statistics. 5) Why the other options are incorrect - "Northern Suburbs is left skewed": incorrect — the ordering of mean, median, mode shows mean > median > mode, which indicates right (positive) skew, not left. - "Southern Suburbs is more dispersed than Northern Suburbs": incorrect — SD 8 (Southern) is less than SD 10 (Northern), so Southern is less dispersed. - "The mean is a good measure of central tendency in Northern Suburbs": incorrect because the Northern distribution is skewed (mean pulled above median), so the median is generally a more robust measure of central tendency there. 6) Conclusion / confirmation - The correct statement is: "The distribution of training hours in Southern Suburbs is symmetrical." This matches the provided answer.

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