{"id":1009,"date":"2026-01-06T13:01:14","date_gmt":"2026-01-06T03:01:14","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1009"},"modified":"2026-01-07T22:27:48","modified_gmt":"2026-01-07T12:27:48","slug":"year11-math-2-2-6-warm-up-questions-univariate-data-analysis-the-power-of-prediction","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1009","title":{"rendered":"Year11 MATH 2-2-6 Warm-up Questions-Univariate Data Analysis (The Power of Prediction)"},"content":{"rendered":"\n

In Chapter 6<\/strong>, we move from simply “looking at numbers” to interpreting the story<\/strong> 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\u2019s about Standard Deviation<\/strong>, the Normal Distribution<\/strong>, and using z<\/mi>z<\/annotation><\/semantics><\/math>-scores<\/strong> to compare data from completely different worlds.<\/p>\n\n\n\n

Concepts and Skills Covered:<\/strong><\/h3>\n\n\n\n
    \n
  1. Measures of Central Tendency:<\/strong> Understanding when the median is a better “middle” than the mean (skewed data).<\/li>\n\n\n\n
  2. Measures of Spread:<\/strong> Mastering Range, Interquartile Range (IQR), and Standard Deviation (\u03c3<\/mi>\\sigma<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n
  3. The <\/strong>1.5<\/mn>\u00d7<\/mo>I<\/mi>Q<\/mi>R<\/mi><\/mrow>1.5 \\times IQR<\/annotation><\/semantics><\/math> Rule:<\/strong> Mathematically identifying outliers.<\/li>\n\n\n\n
  4. The Normal Distribution:<\/strong> Applying the 68\u201395\u201399.7% rule<\/strong>.<\/li>\n\n\n\n
  5. Standardized Scores (<\/strong>z<\/mi>z<\/annotation><\/semantics><\/math>-scores):<\/strong> Using z<\/mi>=<\/mo>x<\/mi>\u2212<\/mo>\u03bc<\/mi><\/mrow>\u03c3<\/mi><\/mfrac><\/mrow>z = \\frac{x – \\mu}{\\sigma}<\/annotation><\/semantics><\/math> to compare relative performance.<\/li>\n<\/ol>\n\n\n\n

    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?<\/p>\n\n\n\n

    A. The Range<\/p>\n\n\n\n

    B. The Mean<\/p>\n\n\n\n

    C. The Median<\/p>\n\n\n\n

    D. The Mode<\/p>\n\n\n\n

    Hint: Think about which average gets ‘dragged’ by a few very large numbers.<\/p>\n\n\n\n

    Q2. In a dataset with Q<\/em>1\u200b=15 and Q<\/em>3\u200b=35, what is the lower boundary for identifying an outlier using the standard 1.5\u00d7IQR<\/em> rule?<\/p>\n\n\n\n

    A. 15<\/p>\n\n\n\n

    B. 0<\/p>\n\n\n\n

    C. \u221215<\/p>\n\n\n\n

    D. 5<\/p>\n\n\n\n

    Hint: Calculate the Interquartile Range (I<\/em>QR<\/em>) first, then multiply it by 1.5.<\/p>\n\n\n\n

    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?<\/p>\n\n\n\n

    A. Class B has more students.<\/p>\n\n\n\n

    B. Class B has a higher average score.<\/p>\n\n\n\n

    C. Class A has a higher range.<\/p>\n\n\n\n

    D. Class A’s results are more consistent.<\/p>\n\n\n\n

    Hint: Standard deviation measures how ‘spread out’ the scores are from the middle.<\/p>\n\n\n\n

    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?<\/p>\n\n\n\n

    A. 68%<\/p>\n\n\n\n

    B. 50%<\/p>\n\n\n\n

    C. 95%<\/p>\n\n\n\n

    D. 34%<\/p>\n\n\n\n

    Hint: How many standard deviations away from the mean are 65 and 75?<\/p>\n\n\n\n

    Q5. A student scored 85 on a test where the mean was 70 and the standard deviation was 10. What is their z<\/em>-score?<\/p>\n\n\n\n

    A. \u22121.5<\/p>\n\n\n\n

    B. 1.5<\/p>\n\n\n\n

    C. 15<\/p>\n\n\n\n

    D. 1.0<\/p>\n\n\n\n

    Hint: Use the formula z<\/em>=\u03c3<\/em>x<\/em>\u2212\u03bc<\/em>\u200b.<\/p>\n\n\n\n

    Q6. In a box plot, what does the ‘box’ itself represent?<\/p>\n\n\n\n

    A. The standard deviation.<\/p>\n\n\n\n

    B. The entire range of the data.<\/p>\n\n\n\n

    C. The mean of the data.<\/p>\n\n\n\n

    D. The middle 50% of the data.<\/p>\n\n\n\n

    Hint: The box is defined by the values of Q<\/em>1\u200b and Q<\/em>3\u200b.<\/p>\n\n\n\n

    Q7. If every value in a dataset is increased by 10, how does the standard deviation change?<\/p>\n\n\n\n

    A. It increases by 10\u200b.<\/p>\n\n\n\n

    B. It increases by 10.<\/p>\n\n\n\n

    C. It becomes 10 times larger.<\/p>\n\n\n\n

    D. It remains the same.<\/p>\n\n\n\n

    Hint: Think about whether the data points are further apart from each other after the increase.<\/p>\n\n\n\n

    Q8. Which z<\/em>-score indicates a better relative performance: z<\/em>=1.2 in Math or z<\/em>=1.5 in English?<\/p>\n\n\n\n

    A. They are the same.<\/p>\n\n\n\n

    B. It depends on the raw marks.<\/p>\n\n\n\n

    C. z<\/em>=1.2 in Math<\/p>\n\n\n\n

    D. z<\/em>=1.5 in English<\/p>\n\n\n\n

    Hint: A z<\/em>-score tells you how many standard deviations you are from the mean.<\/p>\n\n\n\n

    Q9. In a normal distribution, what percentage of data falls above a z<\/em>-score of 0?<\/p>\n\n\n\n

    A. 50%<\/p>\n\n\n\n

    B. 34%<\/p>\n\n\n\n

    C. 68%<\/p>\n\n\n\n

    D. 100%<\/p>\n\n\n\n

    Hint: A z<\/em>-score of 0 represents the exact middle of the curve.<\/p>\n\n\n\n

    Q10. Which of the following is most affected by a single extreme outlier?<\/p>\n\n\n\n

    A. Mode<\/p>\n\n\n\n

    B. Median<\/p>\n\n\n\n

    C. Range<\/p>\n\n\n\n

    D. Interquartile Range (IQR)<\/p>\n\n\n\n

    Hint: Think about which calculation relies entirely on the ‘edges’ of the data.<\/p>\n\n\n\n

    <\/p>\n\n\n\n

    You have now completed the warm-up for all six chapters<\/strong> of Year 11 General Mathematics! You have a strong conceptual grasp of:<\/p>\n\n\n\n

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

      Since we have finished the “Textbook” phase for General Mathematics, would you like to:<\/strong><\/p>\n\n\n\n

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

        *******************<\/p>\n\n\n\n

        You should be proud of the ground we covered today. You\u2019ve successfully built the mental “textbook” for General Mathematics<\/strong>, moving from complex financial models to the logic of matrices and the power of statistical prediction.<\/p>\n\n\n\n

        Before you head off, I’ll leave you with a final “Grade A” Insight<\/strong> for your future General Math assessment.<\/p>\n\n\n\n

        \n\n\n\n

        \ud83d\udca1 The Pro-Tip: Mastering the “Strength” of a Trend<\/h3>\n\n\n\n

        In General Mathematics, particularly in the PSMT (Data Investigation)<\/strong>, you aren’t just asked to draw a line through dots. To get the highest marks, you must use the Coefficient of Determination (<\/strong>r<\/mi>2<\/mn><\/msup>r^2<\/annotation><\/semantics><\/math>)<\/strong>.<\/p>\n\n\n\n
          \n
        • The “C” Student:<\/strong> Describes a graph as “looking like it has a strong relationship.”<\/li>\n\n\n\n
        • The “A” Student:<\/strong> Calculates the r<\/mi>2<\/mn><\/msup>r^2<\/annotation><\/semantics><\/math> value on their graphics calculator. If r<\/mi>2<\/mn><\/msup>r^2<\/annotation><\/semantics><\/math> = 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.”<\/em><\/li>\n<\/ul>\n\n\n\n
          \n\n\n\n

          Your Progress Summary:<\/h3>\n\n\n\n
            \n
          • [x] Essential Mathematics (Applied):<\/strong> Textbook & Warm-ups complete.<\/li>\n\n\n\n
          • [x] General Mathematics (General):<\/strong> Textbook & Warm-ups complete.<\/li>\n\n\n\n
          • [ ] Mathematical Methods (ATAR):<\/strong> Loading…<\/em><\/li>\n<\/ul>\n\n\n\n

            When you’re back tomorrow, we can tackle the “heavyweight” of the QCAA math subjects.<\/p>\n\n\n\n

            ********************************<\/p>\n\n\n\n

            \u7b2c6\u7ae0\u3067\u306f\u3001\u5358\u306b\u300c\u6570\u5b57\u3092\u898b\u308b\u300d\u3060\u3051\u3067\u306a\u304f\u3001\u6570\u5b57\u304c\u4f1d\u3048\u308b\u30b9\u30c8\u30fc\u30ea\u30fc\u3092\u89e3\u91c8\u3059\u308b\u6bb5\u968e\u3078\u3068\u9032\u307f\u307e\u3059\u3002\u5358\u5909\u91cf\u30c7\u30fc\u30bf\u5206\u6790\u3068\u306f\u3001\u5358\u4e00\u306e\u5909\u6570\u306e\u5f62\u72b6\u3001\u4e2d\u5fc3\u3001\u305d\u3057\u3066\u5e83\u304c\u308a\u3092\u7406\u89e3\u3059\u308b\u3053\u3068\u3067\u3059\u3002\u6210\u7e3e\u512a\u79c0\u306a\u65b9\u306b\u3068\u3063\u3066\u3001\u3053\u306e\u7ae0\u306f\u5e73\u5747\u5024\u4ee5\u4e0a\u306e\u3082\u306e\u3001\u3064\u307e\u308a\u6a19\u6e96\u504f\u5dee\u3001\u6b63\u898f\u5206\u5e03\u3001\u305d\u3057\u3066z<\/mi>z<\/annotation><\/semantics><\/math>\u30b9\u30b3\u30a2\u3092\u7528\u3044\u305f\u5168\u304f\u7570\u306a\u308b\u4e16\u754c\u306e\u30c7\u30fc\u30bf\u306e\u6bd4\u8f03\u306b\u3064\u3044\u3066\u5b66\u3073\u307e\u3059\u3002<\/p>\n\n\n\n

            \u5b66\u7fd2\u5185\u5bb9\u3068\u30b9\u30ad\u30eb\uff1a<\/p>\n\n\n\n

              \n
            1. \u4e2d\u5fc3\u50be\u5411\u306e\u5c3a\u5ea6\uff1a\u4e2d\u592e\u5024\u304c\u5e73\u5747\u5024\u3088\u308a\u3082\u300c\u4e2d\u5fc3\u300d\u3068\u3057\u3066\u9069\u5207\u3067\u3042\u308b\u5834\u5408\uff08\u6b6a\u3093\u3060\u30c7\u30fc\u30bf\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<\/li>\n\n\n\n
            2. \u5e83\u304c\u308a\u306e\u5c3a\u5ea6\uff1a\u7bc4\u56f2\u3001\u56db\u5206\u4f4d\u7bc4\u56f2\uff08IQR\uff09\u3001\u6a19\u6e96\u504f\u5dee\uff08\u03c3<\/mi>\\sigma<\/annotation><\/semantics><\/math>\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<\/li>\n\n\n\n
            3. 1.5<\/mn>\u00d7<\/mo>I<\/mi>Q<\/mi>R<\/mi><\/mrow>1.5 \\times IQR<\/annotation><\/semantics><\/math>\u30eb\u30fc\u30eb\uff1a\u5916\u308c\u5024\u3092\u6570\u5b66\u7684\u306b\u7279\u5b9a\u3059\u308b\u3002<\/li>\n\n\n\n
            4. \u6b63\u898f\u5206\u5e03\uff1a68%\u201395%\u201399.7%\u30eb\u30fc\u30eb\u3092\u9069\u7528\u3059\u308b\u3002<\/li>\n\n\n\n
            5. \u6a19\u6e96\u5316\u30b9\u30b3\u30a2\uff08z<\/mi>z<\/annotation><\/semantics><\/math>\u30b9\u30b3\u30a2\uff09\uff1az<\/mi>=<\/mo>x<\/mi>\u2212<\/mo>\u03bc<\/mi><\/mrow>\u03c3<\/mi><\/mfrac><\/mrow>z = \\frac{x – \\mu}{\\sigma}<\/annotation><\/semantics><\/math>\u3092\u4f7f\u7528\u3057\u3066\u76f8\u5bfe\u7684\u306a\u30d1\u30d5\u30a9\u30fc\u30de\u30f3\u30b9\u3092\u6bd4\u8f03\u3057\u307e\u3059\u3002<\/li>\n<\/ol>\n\n\n\n

              ***********************************<\/p>\n\n\n\n

              11\u5e74\u751f\u4e00\u822c\u6570\u5b66\u306e\u51686\u7ae0\u306e\u30a6\u30a9\u30fc\u30e0\u30a2\u30c3\u30d7\u304c\u5b8c\u4e86\u3057\u307e\u3057\u305f\uff01\u4ee5\u4e0b\u306e\u6982\u5ff5\u3092\u3057\u3063\u304b\u308a\u3068\u7406\u89e3\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

              \u91d1\u878d\uff08\u91d1\u5229\u3068\u30a4\u30f3\u30d5\u30ec\uff09
              \u6e2c\u5b9a\uff083\u6b21\u5143\u56f3\u5f62\u3068\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\uff09
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              \u4e09\u89d2\u6cd5\uff08\u76f4\u89d2\u3067\u306a\u3044\u4e09\u89d2\u5f62\uff09
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              \u7d71\u8a08\uff08\u6b63\u898f\u5206\u5e03\u3068z<\/mi>z<\/annotation><\/semantics><\/math>\u30b9\u30b3\u30a2\uff09<\/p>\n\n\n\n

              \u4e00\u822c\u6570\u5b66\u306e\u300c\u6559\u79d1\u66f8\u300d\u30d5\u30a7\u30fc\u30ba\u306f\u7d42\u4e86\u3057\u307e\u3057\u305f\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u3053\u3068\u3092\u3084\u3063\u3066\u307f\u307e\u305b\u3093\u304b\uff1f<\/p>\n\n\n\n

                \n
              1. \u4e00\u822c\u6570\u5b66\u306b\u7279\u5316\u3057\u305fPSMT\uff08\u554f\u984c\u89e3\u6c7a\u3068\u30e2\u30c7\u30ea\u30f3\u30b0\u8ab2\u984c\uff09\u3092\u4f5c\u6210\u3057\u307e\u3059\u304b\uff1f\uff08\u4f8b\uff1a\u5b9f\u4e16\u754c\u306e\u7d71\u8a08\u3092\u7528\u3044\u305f\u4e8c\u5909\u91cf\u30c7\u30fc\u30bf\u8abf\u67fb\uff09\u3002<\/li>\n\n\n\n
              2. QCAA\u30ac\u30a4\u30c9\u30e9\u30a4\u30f3\u306b\u57fa\u3065\u3044\u30663\u756a\u76ee\u306e\u79d1\u76ee\u306b\u9032\u307f\u307e\u3059\u304b\uff1f\uff08\u4ee3\u6570\u7684\u51e6\u7406\u304c\u6700\u3082\u591a\u3044\u79d1\u76ee\u3067\u3042\u308b\u6570\u5b66\u7684\u624b\u6cd5\u3067\u3059\u3002\uff09<\/li>\n<\/ol>\n\n\n\n

                ********************************<\/p>\n\n\n\n

                \u4eca\u65e5\u5b66\u3093\u3060\u5185\u5bb9\u3092\u8a87\u308a\u306b\u601d\u3046\u3079\u304d\u3067\u3059\u3002\u8907\u96d1\u306a\u91d1\u878d\u30e2\u30c7\u30eb\u304b\u3089\u884c\u5217\u306e\u8ad6\u7406\u3001\u305d\u3057\u3066\u7d71\u8a08\u4e88\u6e2c\u306e\u529b\u3078\u3068\u9032\u307f\u3001\u4e00\u822c\u6570\u5b66\u306e\u6697\u8a18\u300c\u6559\u79d1\u66f8\u300d\u3092\u7121\u4e8b\u306b\u4f5c\u308a\u4e0a\u3052\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

                \u3055\u3066\u3001\u51fa\u767a\u306e\u524d\u306b\u3001\u4eca\u5f8c\u306e\u4e00\u822c\u6570\u5b66\u306e\u8a66\u9a13\u306b\u5411\u3051\u3066\u3001\u6700\u5f8c\u306b\u300cA\u300d\u306e\u8a55\u4fa1\u3068\u306a\u308b\u30d2\u30f3\u30c8\u3092\u304a\u4f1d\u3048\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                \ud83d\udca1 \u30d7\u30ed\u306e\u30a2\u30c9\u30d0\u30a4\u30b9\uff1a\u30c8\u30ec\u30f3\u30c9\u306e\u300c\u5f37\u3055\u300d\u3092\u30de\u30b9\u30bf\u30fc\u3059\u308b<\/p>\n\n\n\n

                \u4e00\u822c\u6570\u5b66\u3001\u7279\u306bPSMT\uff08\u30c7\u30fc\u30bf\u8abf\u67fb\uff09\u3067\u306f\u3001\u70b9\u306b\u7dda\u3092\u5f15\u304f\u3060\u3051\u3067\u306f\u5341\u5206\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u6700\u9ad8\u70b9\u3092\u53d6\u308b\u306b\u306f\u3001\u6c7a\u5b9a\u4fc2\u6570\uff08r<\/mi>2<\/mn><\/msup>r^2<\/annotation><\/semantics><\/math>\uff09\u3092\u4f7f\u3046\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

                \u300cC\u300d\u306e\u751f\u5f92\uff1a\u30b0\u30e9\u30d5\u3092\u300c\u5f37\u3044\u95a2\u4fc2\u304c\u3042\u308b\u3088\u3046\u306b\u898b\u3048\u308b\u300d\u3068\u8868\u73fe\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

                \u300cA\u300d\u306e\u751f\u5f92\uff1a\u30b0\u30e9\u30d5\u96fb\u5353\u3067r<\/mi>2<\/mn><\/msup>r^2<\/annotation><\/semantics><\/math>\u306e\u5024\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002 r<\/mi>2<\/mn><\/msup>=<\/mo>0.85<\/mn><\/mrow>r^2 = 0.85<\/annotation><\/semantics><\/math> \u306e\u5834\u5408\u3001\u300c\u5f93\u5c5e\u5909\u6570\uff08\u4f8b\uff1a\u71c3\u6599\u6d88\u8cbb\u91cf\uff09\u306e\u5909\u52d5\u306e 85% \u306f\u72ec\u7acb\u5909\u6570\uff08\u4f8b\uff1a\u901f\u5ea6\uff09\u306e\u5909\u52d5\u306b\u3088\u3063\u3066\u8aac\u660e\u3067\u304d\u3001\u975e\u5e38\u306b\u5f37\u3044\u7dda\u5f62\u8fd1\u4f3c\u3092\u793a\u3057\u3066\u3044\u308b\u300d\u3068\u8a18\u8ff0\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

                \u9032\u6357\u72b6\u6cc1\u306e\u6982\u8981\uff1a
                [x] \u57fa\u790e\u6570\u5b66\uff08\u5fdc\u7528\uff09\uff1a\u6559\u79d1\u66f8\u3068\u30a6\u30a9\u30fc\u30e0\u30a2\u30c3\u30d7\u3092\u5b8c\u4e86\u3057\u307e\u3057\u305f\u3002
                [x] \u4e00\u822c\u6570\u5b66\uff08\u4e00\u822c\uff09\uff1a\u6559\u79d1\u66f8\u3068\u30a6\u30a9\u30fc\u30e0\u30a2\u30c3\u30d7\u3092\u5b8c\u4e86\u3057\u307e\u3057\u305f\u3002
                [] \u6570\u5b66\u7684\u624b\u6cd5\uff08ATAR\uff09\uff1a\u8aad\u307f\u8fbc\u307f\u4e2d\u2026\u3050\u3063\u3059\u308a\u304a\u4f11\u307f\u304f\u3060\u3055\u3044\uff01<\/p>\n\n\n\n

                \u660e\u65e5\u623b\u3063\u3066\u304d\u305f\u3089\u3001QCAA \u6570\u5b66\u79d1\u76ee\u306e\u300c\u30d8\u30d3\u30fc\u7d1a\u300d\u306b\u6311\u6226\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

                *************************************************************************<\/p>\n\n\n\n

                Right answer;<\/p>\n\n\n\n

                Q1. The Mean<\/p>\n\n\n\n

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

                Q2. C. \u221215<\/p>\n\n\n\n

                IQR<\/em>=35\u221215=20. The lower boundary is Q<\/em>1\u200b\u2212(1.5\u00d720)=15\u221230=\u221215.<\/p>\n\n\n\n

                Q3. D. Class A’s results are more consistent.<\/p>\n\n\n\n

                A lower standard deviation indicates that the data points are clustered closer to the mean.<\/p>\n\n\n\n

                Q4. A. 68%<\/p>\n\n\n\n

                The range 65<\/mn>65<\/annotation><\/semantics><\/math> to 75<\/mn>75<\/annotation><\/semantics><\/math> is exactly one standard deviation (5<\/mn>5<\/annotation><\/semantics><\/math>) below and above the mean (70<\/mn>70<\/annotation><\/semantics><\/math>). The 68-95-99.7 rule states 68<\/mn>%<\/mi><\/mrow>68\\%<\/annotation><\/semantics><\/math> of data falls within 1<\/mn>\u03c3<\/mi><\/mrow>1\\sigma<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

                Q5. B. 1.5<\/p>\n\n\n\n

                z<\/mi>=<\/mo>(<\/mo>85<\/mn>\u2212<\/mo>70<\/mn>)<\/mo>\/<\/mi>10<\/mn>=<\/mo>15<\/mn>\/<\/mi>10<\/mn>=<\/mo>1.5<\/mn><\/mrow>z = (85 – 70) \/ 10 = 15 \/ 10 = 1.5<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n

                Q6. D. The middle 50<\/mn>%<\/mi><\/mrow>50\\%<\/annotation><\/semantics><\/math> of the data.<\/p>\n\n\n\n

                The box spans from the first quartile (Q<\/mi>1<\/mn><\/msub>Q_1<\/annotation><\/semantics><\/math>) to the third quartile (Q<\/mi>3<\/mn><\/msub>Q_3<\/annotation><\/semantics><\/math>), which contains the interquartile range.<\/p>\n\n\n\n

                Q7. D. It remains the same.<\/p>\n\n\n\n

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

                Q8. D. z<\/em>=1.5 in English<\/p>\n\n\n\n

                A higher z<\/mi>z<\/annotation><\/semantics><\/math>-score means the individual performed further above the mean compared to their peers.<\/p>\n\n\n\n

                Q9. A. 50%<\/p>\n\n\n\n

                A z<\/mi>z<\/annotation><\/semantics><\/math>-score of 0<\/mn>0<\/annotation><\/semantics><\/math> is the mean. In a perfectly symmetrical normal distribution, half the data is above the mean.<\/p>\n\n\n\n

                Q10. C. Range<\/p>\n\n\n\n

                The range is calculated using only the maximum and minimum values. If one of those is an outlier, the entire range changes drastically.<\/p>\n\n\n\n

                ********************<\/p>\n\n\n\n

                <\/p>\n","protected":false},"excerpt":{"rendered":"

                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\u2019s about Standard Deviation, the Normal Distribution, and using zz-scores to compare data […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1009","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1009","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1009"}],"version-history":[{"count":5,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1009\/revisions"}],"predecessor-version":[{"id":1016,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1009\/revisions\/1016"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1009"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1009"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1009"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}