Statistics — Andhra Pradesh (SSC) Class 9 Mathematics Solutions (Free)

Step 1: Definitions: Mean: The average of all values in the data set. Median: The middle value when data is arranged in order. Mode: The value that appears most frequently. Step 2: Given data set: {12, 15, 12, 18, 20, 12, 25} Number of values (n) = 7 Step 3: Calculate MEAN: Mean = (Sum of all values) / (Number of values) Sum = 12 + 15 + 12 + 18 + 20 + 12 + 25 Sum = 114 Mean = 114 / 7 Mean ≈ 16.29 Step 4: Calculate MEDIAN: Arrange data in ascending order: {12, 12, 12, 15, 18, 20, 25} Since n = 7 (odd number), median is the middle value at position (n+1)/2 = 4th position. Median = 15 Step 5: Calculate MODE: Frequency count: 12 appears 3 times 15 appears 1 time 18 appears 1 time 20 appears 1 time 25 appears 1 time The value 12 appears most frequently (3 times). Mode = 12 Step 6: Summary: Mean = 114/7 ≈ 16.29 Median = 15 Mode = 12 Answer: Mean ≈ 16.29; Median = 15; Mode = 12.

Step 1: Create frequency distribution table: Class | Midpoint (x) | Frequency (f) | f×x 0-10 | 5 | 5 | 25 10-20 | 15 | 8 | 120 20-30 | 25 | 12 | 300 30-40 | 35 | 7 | 245 40-50 | 45 | 3 | 135 Step 2: Calculate midpoint of each class: Midpoint = (Lower limit + Upper limit) / 2 0-10: (0 + 10)/2 = 5 10-20: (10 + 20)/2 = 15 20-30: (20 + 30)/2 = 25 30-40: (30 + 40)/2 = 35 40-50: (40 + 50)/2 = 45 Step 3: Calculate f × x for each class: 5 × 5 = 25 15 × 8 = 120 25 × 12 = 300 35 × 7 = 245 45 × 3 = 135 Step 4: Calculate totals: Total frequency (Σf) = 5 + 8 + 12 + 7 + 3 = 35 Total of f×x (Σf×x) = 25 + 120 + 300 + 245 + 135 = 825 Step 5: Calculate mean: Mean = Σ(f×x) / Σf Mean = 825 / 35 Mean = 23.57 (approximately) Step 6: The mean lies in the 20-30 class, which makes sense as it has the highest frequency. Answer: Mean of grouped data = 825/35 ≈ 23.57.

Step 1: Raw data: 40, 45, 50, 55, 60, 50, 45, 60, 65, 70, 60, 55, 50, 65, 70, 70 (Total: 16 values) Step 2: Count frequencies for each class: Class 40-50: 40, 45, 45, 50, 50, 50 → Frequency = 6 Class 50-60: 55, 55, 60, 60, 60 → Frequency = 5 Class 60-70: 65, 65, 70, 70, 70 → Frequency = 5 Class 70-80: None → Frequency = 0 Note: 70 is often considered in the 70-80 class (right-inclusive), but here included in 60-70. Correction: Let's use left-inclusive intervals. Class 40-50: 40, 45, 45 → Frequency = 3 Class 50-60: 50, 50, 50, 55, 55 → Frequency = 5 Class 60-70: 60, 60, 60, 65, 65 → Frequency = 5 Class 70-80: 70, 70, 70 → Frequency = 3 Total: 3 + 5 + 5 + 3 = 16 ✓ Step 3: Frequency Distribution Table: Class | Frequency | Relative Frequency 40-50 | 3 | 3/16 = 0.1875 50-60 | 5 | 5/16 = 0.3125 60-70 | 5 | 5/16 = 0.3125 70-80 | 3 | 3/16 = 0.1875 Step 4: Bar Graph Description: X-axis: Class intervals (40-50, 50-60, 60-70, 70-80) Y-axis: Frequency (scale 0-5) Bar heights: Class 40-50: bar height = 3 Class 50-60: bar height = 5 Class 60-70: bar height = 5 Class 70-80: bar height = 3 Step 5: Observations: - Maximum frequency in classes 50-60 and 60-70 (5 each) - Symmetric distribution around classes 50-70 - The data is concentrated in the middle ranges (50-70) Answer: Frequency distribution table created above. Bar graph shows bars of heights 3, 5, 5, 3 for classes 40-50, 50-60, 60-70, 70-80 respectively.

Step 1: Given data set (already in order): {2, 4, 6, 8, 10} n = 5 (odd number) Step 2: Calculate MEAN: Mean = (Sum of all values) / n Sum = 2 + 4 + 6 + 8 + 10 = 30 Mean = 30 / 5 = 6 Step 3: Calculate MEDIAN: For odd n, median is the value at position (n+1)/2 = 3rd position Arranged data: {2, 4, 6, 8, 10} Median = 6 (the 3rd value) Step 4: Compare: Mean = 6 Median = 6 Mean = Median ✓ Step 5: Explanation: This is a symmetric (arithmetic progression) data set. The differences between consecutive terms are equal: 4-2=2, 6-4=2, 8-6=2, 10-8=2 In symmetric distributions, the mean and median are equal. The middle value (6) is also the average of all values. Step 6: Mode: No value repeats in this data set. Mode = No mode (or all values are equally frequent) Answer: Mean = 6, Median = 6. For this symmetric data set, mean equals median.

Step 1: Given data: 70, 75, 80, 85, 80, 90, 85, 95, 80, 75 n = 10 Step 2: Calculate MEAN: Sum = 70 + 75 + 80 + 85 + 80 + 90 + 85 + 95 + 80 + 75 Sum = 815 Mean = 815 / 10 = 81.5 Step 3: Calculate MEDIAN: Arrange in ascending order: {70, 75, 75, 80, 80, 80, 85, 85, 90, 95} n = 10 (even), so median = average of 5th and 6th values 5th value = 80 6th value = 80 Median = (80 + 80) / 2 = 80 Step 4: Calculate MODE: Frequency count: 70: 1 time 75: 2 times 80: 3 times 85: 2 times 90: 1 time 95: 1 time Mode = 80 (appears 3 times, most frequent) Step 5: Calculate RANGE: Range = Maximum value - Minimum value Range = 95 - 70 = 25 Step 6: Count students scoring ABOVE the mean: Mean = 81.5 Scores above 81.5: 85, 90, 85, 95 Number of students above mean = 4 Step 7: Summary: Mean = 81.5 Median = 80 Mode = 80 Range = 25 Students above mean = 4 out of 10 = 40% Answer: Mean = 81.5; Median = 80; Mode = 80; Range = 25; 4 students scored above the mean.

Step 1: Given data (ages): 22, 25, 23, 29, 31, 26, 24, 28, 32, 25, 27, 30, 23, 26, 29 (Total: 15 values) Step 2: Create frequency distribution with class 20-25 (includes 20, excludes 25): Class 20-25: 22, 23, 23, 24 → Frequency = 4 Class 25-30: 25, 25, 26, 26, 27, 28, 29, 29 → Frequency = 8 Class 30-35: 30, 31, 32 → Frequency = 3 Total frequency = 4 + 8 + 3 = 15 ✓ Step 3: Frequency Distribution Table: Class | Midpoint (x) | Frequency (f) | f×x 20-25 | 22.5 | 4 | 90 25-30 | 27.5 | 8 | 220 30-35 | 32.5 | 3 | 97.5 Note: Midpoint = (20+25)/2 = 22.5, (25+30)/2 = 27.5, (30+35)/2 = 32.5 Step 4: Calculate f × x: 22.5 × 4 = 90 27.5 × 8 = 220 32.5 × 3 = 97.5 Step 5: Calculate totals: Σf = 4 + 8 + 3 = 15 Σ(f × x) = 90 + 220 + 97.5 = 407.5 Step 6: Calculate mean: Mean = Σ(f × x) / Σf Mean = 407.5 / 15 Mean ≈ 27.17 years Step 7: Verification: The mean falls in the class 25-30, which has the highest frequency (8). This makes sense. Answer: Frequency distribution created. Mean age = 407.5/15 ≈ 27.17 years.

Step 1: Method for finding median with odd number of values: For n values (where n is odd), median is the value at position (n+1)/2 when arranged in ascending order. Step 2: Given data: {3, 7, 5, 9, 1, 8, 4, 6, 2, 10, 11} n = 11 (odd) Step 3: Find position of median: Position = (n + 1) / 2 = (11 + 1) / 2 = 12 / 2 = 6th position Step 4: Arrange data in ascending order: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} Position: 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th Step 5: Identify median: Median = 6th value = 6 Step 6: Verification: Values below median: {1, 2, 3, 4, 5} → 5 values Median: {6} Values above median: {7, 8, 9, 10, 11} → 5 values The median divides the data into two equal halves. ✓ Step 7: Properties: For any data set with odd n, the median always divides the data into two equal parts. Number of values below median = Number of values above median = (n-1)/2 = 5 Answer: Median = 6. The 6th value when arranged in ascending order {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.