Class 11 Statistics — Exercise Solutions
Key Concepts Quick Recap
| Concept | Formula | Notes |
|---|
| Mean (Ungrouped) | x̄ = Σx / N | Affected by extreme values |
| Median (Odd n) | Middle value after arranging | Not affected by outliers |
| Median (Even n) | Average of two middle values | |
| Mode | Most frequent value | May have no mode, one, or multiple |
| Range | Maximum − Minimum | Simplest measure of dispersion |
| Variance (σ²) | Σ(x − x̄)² / N | In squared units |
| Standard Deviation | √Variance | Same units as data |
| Coefficient of Variation | (σ / x̄) × 100 | Relative dispersion (%) |
| Karl Pearson's r | Σ(x−x̄)(y−ȳ) / [Nσₓσᵧ] | Range: −1 to +1 |
I. Choose the Best Answer (MCQs)
1. The sum of deviations of all values from the mean is: (a) Maximum (b) Minimum (c) Zero (d) Infinite
Answer(c) Zero Property of mean: Σ(xᵢ − x̄) = 0 always. This is because the mean is the balancing point of the distribution. Positive and negative deviations cancel out exactly.
2. Which measure of central tendency is not affected by extreme values? (a) Mean (b) Mode (c) Median (d) Range
Answer(c) Median Median: positional average — only depends on middle value(s). Extreme values (outliers) at ends don't affect the middle position. Mean is pulled toward extreme values. Mode is also unaffected by extremes but depends on frequency.
3. If the mean of 5 observations 4, 6, x, 8, 10 is 7, then x = (a) 5 (b) 6 (c) 7 (d) 8
Answer(c) 7 Mean = Σx / N → 7 = (4+6+x+8+10) / 5 → 35 = 28 + x → x = 7.
4. The measure of dispersion which is expressed in the same units as the data is: (a) Variance (b) Mean Deviation (c) Standard Deviation (d) Both (b) and (c)
Answer(d) Both (b) and (c) Variance: expressed in squared units. Mean deviation and standard deviation: both expressed in original units of data. Standard deviation is preferred because it uses all data points and is mathematically tractable.
5. Karl Pearson's coefficient of correlation (r) lies between: (a) 0 and 1 (b) −1 and 0 (c) −1 and +1 (d) −∞ and +∞
Answer(c) −1 and +1 r = −1: perfect negative correlation. r = 0: no linear correlation. r = +1: perfect positive correlation. r between 0.7 and 1.0: strong positive correlation. r between −1.0 and −0.7: strong negative correlation.
6. Primary data is collected: (a) From published sources (b) First-hand, directly for the specific purpose of investigation (c) From secondary published reports (d) From internet sources only
Answer(b) First-hand, directly for the specific purpose of investigation Primary data: original data collected by investigator for the first time for a specific purpose. Methods: questionnaire, interview, observation, experiment. Secondary data: already collected and published by someone else (census, government reports, journals).
7. The coefficient of variation is used to compare: (a) Two datasets with same mean (b) Variability relative to the mean, especially when means differ (c) Two datasets with same standard deviation (d) Only qualitative data
Answer(b) Variability relative to the mean, especially when means differ CV = (σ/x̄) × 100. Used to compare relative dispersion of two or more datasets when units or means differ. Lower CV → more consistent (less variable) data.
II. Short Answer Questions
Q1. Explain the different methods of data collection.
Answer
Data collection methods:
1. Direct Personal Interview: Investigator meets respondents face-to-face. Advantages: personal rapport, clarify doubts, high response rate, can observe non-verbal cues. Disadvantage: expensive, time-consuming, interviewer bias.
2. Indirect Oral Interview: Investigator asks questions through a third party who is close to the subject. Used when direct access is difficult (e.g., census-type surveys in remote areas).
3. Information from Local Sources/Correspondents: Trained local correspondents collect and report data. Used by newspapers, government departments. Cost-effective but may lack accuracy.
4. Questionnaire Sent through Mail/Internet: Printed or online questionnaire sent to respondents. Advantages: wide geographic coverage, anonymous (reduces bias), economical. Disadvantage: low response rate, can't clarify doubts, requires literacy.
5. Census Method: Every unit of the population is observed. Complete and accurate data but very expensive and time-consuming. Example: Census of India (every 10 years).
6. Sampling Method: Only a representative subset (sample) is studied. More practical, faster, less expensive. Requires careful sampling to avoid bias.
Q2. Calculate the mean, median, and mode for the dataset: 5, 7, 8, 9, 7, 6, 7, 10, 5, 6.
Answer
Data arranged in order: 5, 5, 6, 6, 7, 7, 7, 8, 9, 10 (n = 10)
Mean: x̄ = (5+5+6+6+7+7+7+8+9+10) / 10 = 70 / 10 = 7
Median: n = 10 (even) → average of 5th and 6th values. 5th value = 7, 6th value = 7 → Median = (7+7)/2 = 7
Mode: 7 appears 3 times (most frequent) → Mode = 7
In this dataset, Mean = Median = Mode = 7 → approximately normal distribution.
Exam Checklist
- Mean = Σx/N; Σ(x − x̄) = 0 always.
- Median: middle value; not affected by extremes.
- Mode: most frequent; best for categorical data.
- Range = Max − Min; SD = √Variance (in same units as data).
- CV = (σ/x̄) × 100; compares relative dispersion.
- Pearson's r: −1 to +1; 0 = no correlation; ±1 = perfect.
- Primary data: first-hand; Secondary data: already published.
- Census: all units; Sampling: representative subset.
