Statistics — Measures of Central Tendency

Solution:

Measures of central tendency (mean, median, mode) do not deal with spread or extremes — they answer the question: what single value best represents the center or typical observation in the data? They allow us to summarize a large dataset with one number that tells the "story of the center."

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The arithmetic mean is computed by summing all values and dividing by their count. It represents the "balance point" — if the data were weights on a number line, the mean is where the line balances. Because every value contributes to the sum, the mean is sensitive to outliers.

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To find the median, sort the data from smallest to largest, then locate the middle value. The median depends on the position of values, not their exact magnitude — making it resistant to extreme values.

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The mode is the value with the highest frequency — the value that occurs most often. It is especially useful when we care about what is most common: the best-selling size, the most common number of siblings, the most frequent color.

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The mean equals the sum of all values divided by their count. A very large or very small value pulls the total sum up or down, shifting the mean. This is why in the presence of outliers it is often better to also examine the median.

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The median is determined by the middle position in a sorted list, not the numerical size of each value. Even if one value is extremely large, it simply sits at the far end — without changing who is in the middle. The median is therefore not "dragged" by extreme values.

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The mode is the most common value, making it ideal for categorical data: favorite color, shoe brand, preferred drink, etc. In these cases computing a numerical mean is not possible, but it is useful to know which value appears most often.

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The student may have scored very highly on some exams but very poorly on others. The mean hides the spread and possible inconsistency. To truly understand performance, one must also examine spread measures (e.g., median, variance, range) — not the mean alone.

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A small number of very high earners can inflate the mean, making it unrepresentative. The median reports what the employee in the middle earns — a more accurate picture for the majority of workers.

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If every value in the list appears exactly once, no single value is "more common" than the others — therefore there is no mode. Central tendency measures depend on how the data is distributed.

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Three values are close together (10) and one is far away (90). The mean rises significantly due to 90, but both the mode (10) and the median (10) accurately represent most of the data. In this case the median and mode are more appropriate descriptors of the center.

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The data is perfectly symmetric. Mean = (4+5+6+7+8)/5 = 6, median = 6 (middle value). In a symmetric distribution, all three measures of central tendency coincide.

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The median depends on the middle position, so the values must be in order. Without sorting, the concept of "middle" is meaningless.

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A large value added to the far end of the sorted list does not always shift who is in the middle. The median stays stable — especially in large datasets. This illustrates the medians resistance to extreme values.

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The mean is a mathematical result and need not match any original data point. For example, the mean of 1 and 3 is 2 — even though 2 does not appear in the list.

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With an even number of values there is no single middle. We take the two central values (after sorting) and compute their average to produce one central representative value.

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If two or more values appear with the same maximum frequency, the mode is not unique — and the picture becomes more complex. The mode alone is then insufficient to describe the center.

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The value 100 pulls the mean substantially upward, but the median is based on the middle: (3 + 4) / 2 = 3.5. This gap between mean and median signals an extreme value in the dataset.

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The mean reflects value balance, the median reflects the actual middle, and the mode reflects the most common value. Examining all three reveals skewness, outliers, and data concentration around specific values.

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Statistics is about understanding reality through numbers. When you understand what mean, median, and mode represent, you can choose the right measure for each situation and interpret results correctly — not just mechanically. Conceptual understanding prevents interpretation errors.

Solution:

Sum: 10 + 12 + 14 + 16 = 52.
Mean = 52 ÷ 4 = 13.

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Sum: 80 + 90 + 100 = 270.
Mean = 270 ÷ 3 = 90.

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Sum: 5 + 7 + 7 + 9 = 28.
Mean = 28 ÷ 4 = 7.

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Sum: 60 + 70 + 85 + 95 = 310.
Mean = 310 ÷ 4 = 77.5.

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Sum: 13 + 15 + 17 + 19 = 64.
Mean = 64 ÷ 4 = 16.

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Sum: 100 + 0 = 100.
Mean = 100 ÷ 2 = 50.

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Sum: 20 + 20 + 20 + 21 = 81.
Mean = 81 ÷ 4 = 20.25.

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Sum: 9 + 10 + 11 + 12 = 42.
Mean = 42 ÷ 4 = 10.5.

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Sum: 4 + 10 = 14.
Mean = 14 ÷ 2 = 7.

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Sum: 50 + 55 + 60 + 65 + 70 = 300.
Mean = 300 ÷ 5 = 60. Because the data is symmetric, the mean equals the median.

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Sort: 2, 3, 5, 7, 8. The middle (3rd) value is 5.

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Sort: 4, 6, 8, 10. With 4 values (even count), median = (6 + 8) ÷ 2 = 7.

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Already sorted: 1, 2, 2, 3, 100. The middle (3rd) value is 2. The outlier 100 has no effect on the median.

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The two middle values are both 5. Median = (5 + 5) ÷ 2 = 5.

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Median = (2 + 9) ÷ 2 = 5.5.

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The value 2 appears twice; all others appear once. Mode = 2.

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Both 5 and 7 appear twice; no other value appears more often. Both 5 and 7 are modes — a bimodal distribution.

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Every value appears exactly once — no value is "most common." There is no mode.

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Sum: 1+2+3+4+5 = 15 → Mean = 15 ÷ 5 = 3. The middle (3rd) value = 3 — the median. The distribution is symmetric, so mean = median.

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Mean = (2+2+3+10) ÷ 4 = 17 ÷ 4 = 4.25. Mode = 2 (appears twice). The mean is inflated by 10; the mode reflects the most frequent value.

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Sum: 6 + 6 + 8 + 10 = 30.
Mean = 30 ÷ 4 = 7.5.

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Already sorted. With 6 values, the two middle values are 9 and 16. Median = (9 + 16) ÷ 2 = 25 ÷ 2 = 12.5.

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95 appears three times, 85 appears twice, all others less. Mode = 95.

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The mean is inflated by 30,000 and does not represent most employees. The median (8,000) reflects the middle employees actual earnings — a better descriptor of the typical salary.

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Sum: 3+3+3+3+9 = 21. Mean = 21 ÷ 5 = 4.2. The single high value (9) pulls the mean above 3, even though four out of five values are 3.

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Already sorted. With 7 values, the middle (4th) value is 40.

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Values 2, 3, and 4 each appear twice; no value appears more. All three are modes — a multimodal distribution.

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100 is far from the other values and causes the mean to rise substantially. The median (9) and mode are barely affected. The mean is the most sensitive measure.

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Adding 100 greatly increases the sum and therefore the mean. This is the typical effect of an extreme (outlier) value.

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Adding 100 shifts the middle position slightly, but the median stays in the region of the original values — not pulled to the extreme the way the mean is. This demonstrates the medians resistance to outliers.

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Sum: 2+4+6+8+10+12 = 42. Mean = 42 ÷ 6 = 7.

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Already sorted. Two middle values: 6 and 8. Median = (6 + 8) ÷ 2 = 7. Here mean = median.

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The value 1 appears twice; all others appear once. Mode = 1.

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Two middle values: 30 and 40. Median = (30 + 40) ÷ 2 = 35. The high value 100 has no direct effect on the median.

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Sum: 5 × 5 = 25. Mean = 25 ÷ 5 = 5. In a uniform series, mean = median = mode = 5.

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Mean = (2+4+4+4+10) ÷ 5 = 24 ÷ 5 = 4.8. Mode = 4 (appears 3 times).

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100 is far from the rest and pulls the mean upward. The median (9) better reflects what most of the data looks like.

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60 appears three times; all other scores appear once. Mode = 60 — the score most students received.

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Sum: 2+3+3+4+4+4+5 = 25. Mean = 25 ÷ 7 ≈ 3.57. Middle (4th) value = 4. The median is slightly higher than the mean.

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Median: 3rd value = 2. Mode: 2 appears twice → 2. Mean: pulled upward by 100 → mean > 2. A classic example of an outliers effect: all three measures tell different stories.