Statistics — Telangana (SSC) Class 9 Mathematics Solutions (Free)

Mean: the sum of all values divided by the number of values (the "average"). Median: the middle value when the data is arranged in ascending order. Mode: the value that occurs most frequently in the data set. Given data: 14, 17, 14, 21, 23, 14, 28 (7 values) Mean: Sum = 14 + 17 + 14 + 21 + 23 + 14 + 28 = 131 Mean = 131 / 7 ≈ 18.71 Median: Arrange in order: 14, 14, 14, 17, 21, 23, 28 With 7 values (odd), the median is the middle (4th) value: 17 Mode: 14 appears three times — more often than any other value. Mode = 14

For grouped data, the mean is calculated using the midpoint (x) of each class interval, weighted by its frequency (f): Mean = Σ(f × x) / Σf | Class | Midpoint (x) | Frequency (f) | f × x | |-------|------|------|-------| | 0–10 | 5 | 4 | 20 | | 10–20 | 15 | 9 | 135 | | 20–30 | 25 | 14 | 350 | | 30–40 | 35 | 8 | 280 | | 40–50 | 45 | 5 | 225 | Σf = 4 + 9 + 14 + 8 + 5 = 40 Σ(f × x) = 20 + 135 + 350 + 280 + 225 = 1010 Mean = 1010 / 40 = 25.25 The mean of the grouped data is 25.25.

Sorting the 16 recorded marks into the given class intervals (each interval includes its lower bound up to, but not including, its upper bound): 40–50: 42, 48, 48 → frequency 3 50–60: 51, 58, 51, 58, 51 → frequency 5 60–70: 63, 63, 68, 63, 68 → frequency 5 70–80: 72, 72, 72 → frequency 3 | Class Interval | Frequency | |---|---| | 40–50 | 3 | | 50–60 | 5 | | 60–70 | 5 | | 70–80 | 3 | | **Total** | **16** | A bar graph representing this data would show four bars along the horizontal axis, one per class interval, with heights of 3, 5, 5, and 3 respectively on the vertical (frequency) axis — the two middle bars (50–60 and 60–70) would be the tallest, showing that most students' marks clustered in the middle range.

Given data: 3, 6, 9, 12, 15 (already in ascending order, 5 values) Mean: Sum = 3 + 6 + 9 + 12 + 15 = 45 Mean = 45 / 5 = 9 Median: With 5 values (odd), the median is the middle (3rd) value: 9 Both the mean and median equal 9. Why this happens: this data set is arithmetically evenly spaced (each value is exactly 3 more than the previous one) and symmetric around its middle term — for any such evenly-spaced (arithmetic) sequence, the middle value is always exactly equal to the sequence's average, because the amounts by which values above the middle exceed it are exactly balanced by the amounts values below the middle fall short of it.

Given data: 65, 72, 78, 82, 78, 88, 82, 92, 78, 72 (10 values) Mean: Sum = 65+72+78+82+78+88+82+92+78+72 = 787 Mean = 787/10 = 78.7 Median: Arrange in order: 65, 72, 72, 78, 78, 78, 82, 82, 88, 92 With 10 values (even), median = average of the 5th and 6th values = (78 + 78)/2 = 78 Mode: 78 appears three times — more than any other value. Mode = 78 Range: Range = Highest − Lowest = 92 − 65 = 27 Students scoring above the mean (78.7): 82, 88, 82, 92 — that's 4 students.

Sorting the 15 recorded ages into the given class intervals (each interval includes its lower bound up to, but not including, its upper bound): 20–25: 21, 24, 22, 23, 24, 22 → frequency 6 25–30: 28, 25, 27, 26, 29, 25, 28 → frequency 7 30–35: 33, 34 → frequency 2 | Class Interval | Midpoint (x) | Frequency (f) | f × x | |---|---|---|---| | 20–25 | 22.5 | 6 | 135 | | 25–30 | 27.5 | 7 | 192.5 | | 30–35 | 32.5 | 2 | 65 | Σf = 6 + 7 + 2 = 15 ✓ (matches the 15 original readings) Σ(f × x) = 135 + 192.5 + 65 = 392.5 Mean = Σ(f×x) / Σf = 392.5 / 15 ≈ 26.17 The mean age is approximately 26.17 years.

For a data set with an ODD number of values (n), the median is the single middle value once the data is arranged in ascending order — specifically, the value at position (n + 1)/2. Given 11 values, the median is the value at position (11 + 1)/2 = 6th position after sorting. Arranging the data in ascending order: 12, 21, 23, 34, 43, 45, 56, 67, 78, 89, 90 Counting to the 6th value: 12 (1st), 21 (2nd), 23 (3rd), 34 (4th), 43 (5th), 45 (6th) The median is 45. Unlike the mean, the median is not affected by extreme outlier values (like the 90 in this data set) — it only depends on which value happens to sit in the middle position, which is exactly why the median is often preferred over the mean when a data set contains unusually large or small extreme values.