Master Statistics Quiz - All Sections Combined

1. Calculating Class Width & Limits:

A data set has a minimum value of 45 and a maximum of 132. Construct a frequency distribution with 6 classes. Find the Class Width and the Lower/Upper limits for the first two classes.

Step 1 (Range): Range = Max - Min = 132 - 45 = 87[cite: 43, 762]. Step 2 (Width): Width = Range / #Classes = 87 / 6 = 14.5. Round up to 15[cite: 44, 45, 74]. Step 3 (Limits): Class 1 Lower Limit = 45 (min)[cite: 50, 78]. Class 2 Lower Limit = 45 + 15 = 60[cite: 51, 81]. Class 1 Upper Limit = 60 - 1 = 59[cite: 52, 99]. Results: Class 1: (45-59), Class 2: (60-74).

2. Complete Frequency Distribution Table:

Complete the table for 40 data entries (n=40, width=10). Calculate Midpoint (x), Relative Frequency, and Cumulative Frequency.

Class Limits	Frequency (f)	Midpoint (x)	Relative Frequency	Cumulative Frequency
50 - 59	8	?	?	?
60 - 69	12	?	?	?
70 - 79	10	?	?	?
80 - 89	7	?	?	?
90 - 99	3	?	?	40

Step 1 (Midpoints): (Lower+Upper)/2. Example: (50+59)/2 = 54.5[cite: 118, 659]. Add width (10) for others: 64.5, 74.5, 84.5, 94.5. Step 2 (Relative Frequency): f / n. 8/40 = 0.20, 12/40 = 0.30, 10/40 = 0.25, 7/40 = 0.175, 3/40 = 0.075[cite: 126, 127]. Step 3 (Cumulative Frequency): Summing frequencies: 8, (8+12)=20, (20+10)=30, (30+7)=37, (37+3)=40[cite: 132, 133].

Class	Mid (x)	Rel f	Cum f
50-59	54.5	0.20	8
60-69	64.5	0.30	20
70-79	74.5	0.25	30
80-89	84.5	0.175	37
90-99	94.5	0.075	40

3. Class Boundaries Calculation:

For the class (70 - 79), calculate the Lower and Upper Class Boundaries.

Rule: Subtract 0.5 from Lower Limit and add 0.5 to Upper Limit[cite: 158, 159]. Lower Boundary: 70 - 0.5 = 69.5. Upper Boundary: 79 + 0.5 = 79.5.

4. Ordered Stem-and-Leaf Plot:

Construct an ordered stem-and-leaf plot for the following data: 21, 25, 30, 25, 36, 45, 27.

Step 1 (Stems): Use the tens place: 2, 3, 4[cite: 296]. Step 2 (Leaves): List the ones place for each stem and order them[cite: 336]. 2 | 1 5 5 7 3 | 0 6 4 | 5 Key: 2 | 1 = 21[cite: 311].

5. Pie Chart (Central Angle):

If a category has a relative frequency of 0.35, find the central angle for the pie chart.

Formula: Relative Frequency × 360°[cite: 371]. Calculation: 0.35 × 360 = 126°.

6. Mean, Median, & Outliers:

Data Set: 10, 12, 12, 14, 15, 95

Calculate the Mean and Median. Explain why the Median is a better "typical" measure here.

Mean (x̄): Σx / n = (10+12+12+14+15+95) / 6 = 158 / 6 ≈ 26.3[cite: 482, 603]. Median: Average of the two middle values (12 and 14) = (12+14)/2 = 13[cite: 509, 604]. Explanation: 95 is an outlier[cite: 589]. The Mean is heavily pulled toward the outlier, while the Median is resistant and reflects the center better[cite: 612, 614].

7. Mean of Grouped Data (Table):

Calculate the estimated sample mean from the following frequency distribution.

Class	Midpoint (x)	Frequency (f)	(x · f)
10 - 20	15	4	?
21 - 31	26	6	?
32 - 42	37	10	?
Total	-	n = 20	Σ(x·f) = ?

Step 1 (x · f): (15·4)=60, (26·6)=156, (37·10)=370[cite: 660, 671]. Step 2 (Sum Σ): 60 + 156 + 370 = 586[cite: 660, 671]. Step 3 (Mean x̄): Σ(x·f) / n = 586 / 20 = 29.3[cite: 662, 672].

8. Weighted Mean:

Scores: Quiz (80) weight 30%, Final (90) weight 70%. Calculate the Weighted Mean.

Formula: Σ(x · w) / Σw[cite: 623]. Calculation: (80 · 0.30) + (90 · 0.70) = 24 + 63 = 87[cite: 634, 635].

9. Population Variance Table:

For a population {10, 20, 30, 40} where μ = 25. Complete the table to find σ.

x	Deviation (x - μ)	Squares (x - μ)²
10	-15	225
20	?	?
30	?	?
40	?	?
Σ	0	SSx = ?

Step 1 (Deviations): (20-25)=-5, (30-25)=5, (40-25)=15[cite: 786, 803]. Step 2 (Squares): (-5)²=25, (5)²=25, (15)²=225[cite: 831, 857]. Step 3 (SSx Sum): 225 + 25 + 25 + 225 = 500[cite: 832, 883]. Step 4 (Variance σ²): SSx / N = 500 / 4 = 125[cite: 842, 887]. Step 5 (Std Dev σ): √125 ≈ 11.18[cite: 842, 889].

10. Empirical Rule (68-95-99.7):

A bell-shaped distribution has x̄ = 100 and s = 15. What percentage of data falls between 70 and 130?

Step 1: Distance from mean = (130 - 100) / 15 = 2 standard deviations[cite: 1025]. Rule: Approximately 95% of data falls within 2 standard deviations[cite: 1025, 1028].

11. Five-Number Summary:

Data Set: 5, 7, 12, 15, 18, 22, 30

Find the Five-Number Summary and IQR.

1. Minimum: 5[cite: 1057, 1064]. 2. Q1 (Median of lower half): Median of {5, 7, 12} = 7[cite: 1048, 1049]. 3. Median (Q2): Middle value = 15[cite: 1041, 1047]. 4. Q3 (Median of upper half): Median of {18, 22, 30} = 22[cite: 1042, 1049]. 5. Maximum: 30[cite: 1057, 1064]. IQR: Q3 - Q1 = 22 - 7 = 15[cite: 1053, 1055].

12. Z-Score Calculation:

Value (x) = 85, Mean (μ) = 70, Std Dev (σ) = 5. Calculate and interpret the z-score.

Formula: z = (x - μ) / σ[cite: 1067]. Calculation: (85 - 70) / 5 = 15 / 5 = 3.0[cite: 1067]. Interpretation: The value 85 is 3.0 standard deviations above the mean.

📊 Statistics Chapter 2: Master Practice File

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