問80
頻出度★★☆
横軸を点数（0 ～ 10 点）とし，縦軸を人数とする度数分布のグラフが，次の黒い棒グラフになった場合と，グレーの棒グラフになった場合を考える。二つの棒グラフを比較して言えることはどれか。
| 点数（点） | 黒い棒グラフ（人数） | グレーの棒グラフ（人数） |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 0 |
| 2 | 2 | 0 |
| 3 | 3 | 2 |
| 4 | 4 | 5 |
| 5 | 5 | 11 |
| 6 | 4 | 5 |
| 7 | 3 | 2 |
| 8 | 2 | 0 |
| 9 | 1 | 0 |
| 10 | 0 | 0 |
平成21年度 春期 ITパスポート試験 問80
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 80
Incorrectア
The size relationship is reversed. The gray chart is a narrow distribution concentrated around 5 points (variance 1.04), so its spread is smaller than that of the black chart, which extends from 1 point to 9 points (variance 4). This choice mistakes the eye-catching bar height of 11 people for a large spread; what determines the variance is not the height of the bars but how far the data spreads along the horizontal axis.
- イ
正解 Variance is an index that shows how far each piece of data is spread out from the mean; it is obtained as the average of the squared differences between each value and the mean. Both graphs total 25 people, and both are symmetric about 5 points, so the mean is 5 points in both cases. Comparing how widely the data spreads: the black chart spreads gently over 9 steps from 1 point to 9 points, while the gray chart stays within the 5 steps from 3 points to 7 points and is strongly concentrated at the center, with 11 people at 5 points alone. In other words, the gray chart has fewer values far from the mean (5 points), so its spread is smaller. Calculating the actual values, the variance of black is (1×16 + 2×9 + 3×4 + 4×1 + 5×0 + 4×1 + 3×4 + 2×9 + 1×16) ÷ 25 = 100 ÷ 25 = 4, and the variance of gray is (2×4 + 5×1 + 11×0 + 5×1 + 2×4) ÷ 25 = 26 ÷ 25 = 1.04, so gray (1.04) < black (4). Therefore イ, which states that the variance of the gray bar chart is smaller than that of the black bar chart, is correct.
Incorrectウ
The means are both 5 points and are indeed equal, but that does not mean the variances are equal as well. In the black chart the people are spread over the wide range from 1 point to 9 points, while the gray chart stays within the narrow range from 3 points to 7 points. Calculation gives 4 for black and 1.04 for gray, which are clearly different, so they cannot be said to be equal.
Incorrectエ
A frequency distribution graph lets you read off the number of people (the frequency) for each score directly, so both the mean and the variance can be calculated. As shown above, a numerical variance can in fact be obtained for both charts, so saying that they cannot be compared from this graph alone is wrong.
ここがポイント
- Variance is the average of the squared differences between each value and the mean; the larger the value, the more the data is spread out from the mean. Even when the means are the same, the variances can differ, so check the mean and the variance separately.
- In a frequency distribution graph, the taller and sharper the peak at the center and the narrower its width, the smaller the variance; the wider the tails stretch to the left and right, the larger the variance. The basis for the judgment is not the height of the bars themselves but how far the data extends along the horizontal axis.
用語