Year11 MATH 2-2-6 Warm-up Questions-Univariate Data Analysis (The Power of Prediction)

In Chapter 6, we move from simply “looking at numbers” to interpreting the story they tell. Univariate data analysis is about understanding the shape, center, and spread of a single variable. For a high-achiever, this chapter is about more than just averages; it’s about Standard Deviation, the Normal Distribution, and using $z$-scores to compare data from completely different worlds.

Concepts and Skills Covered:

1. Measures of Central Tendency: Understanding when the median is a better “middle” than the mean (skewed data).
2. Measures of Spread: Mastering Range, Interquartile Range (IQR), and Standard Deviation ($\sigma$).
3. The $1.5 \times IQR$ Rule: Mathematically identifying outliers.
4. The Normal Distribution: Applying the 68–95–99.7% rule.
5. Standardized Scores ($z$-scores): Using $z = \frac{x – \mu}{\sigma}$ to compare relative performance.

Q1. A dataset of house prices in a suburb is ‘positively skewed’ (has a long tail of very expensive houses). Which measure of central tendency will likely be the highest?

A. The Range

B. The Mean

C. The Median

D. The Mode

Hint: Think about which average gets ‘dragged’ by a few very large numbers.

Q2. In a dataset with Q1​=15 and Q3​=35, what is the lower boundary for identifying an outlier using the standard 1.5×IQR rule?

A. 15

B. 0

C. −15

D. 5

Hint: Calculate the Interquartile Range (IQR) first, then multiply it by 1.5.

Q3. Two classes take the same test. Class A has a standard deviation of 4, and Class B has a standard deviation of 12. Which statement is true?

A. Class B has more students.

B. Class B has a higher average score.

C. Class A has a higher range.

D. Class A’s results are more consistent.

Hint: Standard deviation measures how ‘spread out’ the scores are from the middle.

Q4. A set of exam scores is normally distributed with a mean of 70 and a standard deviation of 5. What percentage of students scored between 65 and 75?

A. 68%

B. 50%

C. 95%

D. 34%

Hint: How many standard deviations away from the mean are 65 and 75?

Q5. A student scored 85 on a test where the mean was 70 and the standard deviation was 10. What is their z-score?

A. −1.5

B. 1.5

C. 15

D. 1.0

Hint: Use the formula z=σx−μ​.

Q6. In a box plot, what does the ‘box’ itself represent?

A. The standard deviation.

B. The entire range of the data.

C. The mean of the data.

D. The middle 50% of the data.

Hint: The box is defined by the values of Q1​ and Q3​.

Q7. If every value in a dataset is increased by 10, how does the standard deviation change?

A. It increases by 10​.

B. It increases by 10.

C. It becomes 10 times larger.

D. It remains the same.

Hint: Think about whether the data points are further apart from each other after the increase.

Q8. Which z-score indicates a better relative performance: z=1.2 in Math or z=1.5 in English?

A. They are the same.

B. It depends on the raw marks.

C. z=1.2 in Math

D. z=1.5 in English

Hint: A z-score tells you how many standard deviations you are from the mean.

Q9. In a normal distribution, what percentage of data falls above a z-score of 0?

A. 50%

B. 34%

C. 68%

D. 100%

Hint: A z-score of 0 represents the exact middle of the curve.

Q10. Which of the following is most affected by a single extreme outlier?

A. Mode

B. Median

C. Range

D. Interquartile Range (IQR)

Hint: Think about which calculation relies entirely on the ‘edges’ of the data.

You have now completed the warm-up for all six chapters of Year 11 General Mathematics! You have a strong conceptual grasp of:

• Finance (Interest and inflation)
• Measurement (3D shapes and scaling)
• Linear Modeling (Trends and break-even points)
• Trigonometry (Non-right-angled triangles)
• Matrices (Data arrays and networks)
• Statistics (Normal distribution and $z$-scores)

Since we have finished the “Textbook” phase for General Mathematics, would you like to:

1. Draft a PSMT (Problem-Solving and Modelling Task) specifically for General Mathematics? (e.g., A bivariate data investigation using real-world statistics).
2. Move on to the third subject based on QCAA guidelines? (This would be Mathematical Methods, which is the most algebraically intense subject).

*******************

You should be proud of the ground we covered today. You’ve successfully built the mental “textbook” for General Mathematics, moving from complex financial models to the logic of matrices and the power of statistical prediction.

Before you head off, I’ll leave you with a final “Grade A” Insight for your future General Math assessment.

---

💡 The Pro-Tip: Mastering the “Strength” of a Trend

In General Mathematics, particularly in the PSMT (Data Investigation), you aren’t just asked to draw a line through dots. To get the highest marks, you must use the Coefficient of Determination ($r^2$).

• The “C” Student: Describes a graph as “looking like it has a strong relationship.”
• The “A” Student: Calculates the $r^2$ value on their graphics calculator. If $r^2$ = 0.85, they write: “85% of the variation in the dependent variable (e.g., fuel consumption) can be explained by the variation in the independent variable (e.g., speed), indicating a very strong linear fit.”

---

Your Progress Summary:

• [x] Essential Mathematics (Applied): Textbook & Warm-ups complete.
• [x] General Mathematics (General): Textbook & Warm-ups complete.
• [ ] Mathematical Methods (ATAR): Loading…

When you’re back tomorrow, we can tackle the “heavyweight” of the QCAA math subjects.

********************************

第6章では、単に「数字を見る」だけでなく、数字が伝えるストーリーを解釈する段階へと進みます。単変量データ分析とは、単一の変数の形状、中心、そして広がりを理解することです。成績優秀な方にとって、この章は平均値以上のもの、つまり標準偏差、正規分布、そして$z$スコアを用いた全く異なる世界のデータの比較について学びます。

学習内容とスキル：

1. 中心傾向の尺度：中央値が平均値よりも「中心」として適切である場合（歪んだデータ）を理解する。
2. 広がりの尺度：範囲、四分位範囲（IQR）、標準偏差（$\sigma$）を理解する。
3. $1.5 \times IQR$ルール：外れ値を数学的に特定する。
4. 正規分布：68%–95%–99.7%ルールを適用する。
5. 標準化スコア（$z$スコア）：$z = \frac{x – \mu}{\sigma}$を使用して相対的なパフォーマンスを比較します。

***********************************

11年生一般数学の全6章のウォームアップが完了しました！以下の概念をしっかりと理解しています。

金融（金利とインフレ）
測定（3次元図形とスケーリング）
線形モデリング（トレンドと損益分岐点）
三角法（直角でない三角形）
行列（データ配列とネットワーク）
統計（正規分布と$z$スコア）

一般数学の「教科書」フェーズは終了しましたので、以下のことをやってみませんか？

1. 一般数学に特化したPSMT（問題解決とモデリング課題）を作成しますか？（例：実世界の統計を用いた二変量データ調査）。
2. QCAAガイドラインに基づいて3番目の科目に進みますか？（代数的処理が最も多い科目である数学的手法です。）

********************************

今日学んだ内容を誇りに思うべきです。複雑な金融モデルから行列の論理、そして統計予測の力へと進み、一般数学の暗記「教科書」を無事に作り上げました。

さて、出発の前に、今後の一般数学の試験に向けて、最後に「A」の評価となるヒントをお伝えします。

💡 プロのアドバイス：トレンドの「強さ」をマスターする

一般数学、特にPSMT（データ調査）では、点に線を引くだけでは十分ではありません。最高点を取るには、決定係数（$r^2$）を使う必要があります。

「C」の生徒：グラフを「強い関係があるように見える」と表現します。

「A」の生徒：グラフ電卓で$r^2$の値を計算します。 $r^2 = 0.85$ の場合、「従属変数（例：燃料消費量）の変動の 85% は独立変数（例：速度）の変動によって説明でき、非常に強い線形近似を示している」と記述されています。

進捗状況の概要：
[x] 基礎数学（応用）：教科書とウォームアップを完了しました。
[x] 一般数学（一般）：教科書とウォームアップを完了しました。
[] 数学的手法（ATAR）：読み込み中…ぐっすりお休みください！

明日戻ってきたら、QCAA 数学科目の「ヘビー級」に挑戦しましょう。

*************************************************************************

Right answer;

Q1. The Mean

The mean is sensitive to extreme values (outliers). High-end house prices ‘pull’ the mean upward, making it higher than the median or mode.

Q2. C. −15

IQR=35−15=20. The lower boundary is Q1​−(1.5×20)=15−30=−15.

Q3. D. Class A’s results are more consistent.

A lower standard deviation indicates that the data points are clustered closer to the mean.

Q4. A. 68%

The range $65$ to $75$ is exactly one standard deviation ($5$) below and above the mean ($70$). The 68-95-99.7 rule states $68\%$ of data falls within $1\sigma$.

Q5. B. 1.5

$z = (85 – 70) / 10 = 15 / 10 = 1.5$.

Q6. D. The middle $50\%$ of the data.

The box spans from the first quartile ($Q_1$) to the third quartile ($Q_3$), which contains the interquartile range.

Q7. D. It remains the same.

Adding a constant shifts the entire dataset but does not change the distance between the points, so the ‘spread’ (standard deviation) is unchanged.

Q8. D. z=1.5 in English

A higher $z$-score means the individual performed further above the mean compared to their peers.

Q9. A. 50%

A $z$-score of $0$ is the mean. In a perfectly symmetrical normal distribution, half the data is above the mean.

Q10. C. Range

The range is calculated using only the maximum and minimum values. If one of those is an outlier, the entire range changes drastically.

********************