Questions: Statistics

Q1. Define range and state one limitation of it.

The range of a data set is the difference between its largest and smallest observations. For the scores 10, 30, 45, 60 and 90 the range is 80. The limitation is that it depends on only two values, so a single extreme observation can change it drastically while the rest of the data stay the same. Two data sets with very different spreads inside can have the same range. It also says nothing about how the observations are distributed between the two extremes.

Hinglish: Range = maximum - minimum. Iski limitation yeh hai ki sirf do values par depend karti hai, isliye ek extreme value range ko bahut badal deti hai. Andar ka spread pata nahi chalta.

Q2. The numbers 5, 9, 13, 17 and 21 are given. Calculate their mean deviation about the mean.

The sum is 65 and the mean is 13. The absolute deviations are 8, 4, 0, 4 and 8, whose sum is 24. With five observations the mean deviation about the mean is 24/5 = 4.8. The mean deviation uses absolute values, so it is never negative and it has the same units as the data. Here the typical distance from the mean is 4.8 units.

Hinglish: Mean 13 hai, deviations 8, 4, 0, 4 aur 8 hain. Inka sum 24, isliye MD = 24/5 = 4.8.

Q3. Write the formula for the variance of a frequency distribution.

For values x1 to xn with frequencies f1 to fn and total frequency N equal to the sum of the frequencies, the variance is the sum of f times the squared deviation divided by N. The mean in the formula is the frequency-weighted mean. For a continuous distribution the values are replaced by the mid-points of the classes. The standard deviation is the square root of this variance.

Hinglish: Variance = sum fi (xi - x bar) squared / N, jahan N saari frequencies ka sum hai. Continuous data me xi ki jagah class mid-point lete hain, aur standard deviation iska square root hota hai.

Q4. The variance of 10 observations is 25. Find the standard deviation, and the new variance if every observation is multiplied by 3.

The standard deviation is the square root of the variance, so it is 5. Multiplying every observation by 3 multiplies each deviation by 3 and each squared deviation by 9, so the new variance is 25 times 9, that is 225. The new standard deviation is 15. If a constant were added to every observation instead, the variance would stay 25.

Hinglish: Standard deviation = 5. Har observation ko 3 se multiply karne par variance 9 guna ho jaata hai, isliye naya variance 225 aur naya SD 15. Constant add karne se variance nahi badalta.

Q5. Find the variance of the first four natural numbers.

The numbers are 1, 2, 3 and 4, with sum 10 and mean 2.5. The deviations are minus 1.5, minus 0.5, 0.5 and 1.5, whose squares are 2.25, 0.25, 0.25 and 2.25. The sum of the squares is 5, so the variance is 5/4 = 1.25 and the standard deviation is about 1.12. For the first n natural numbers the general result is variance equal to (n squared - 1)/12, which gives 15/12 = 1.25 for n = 4.

Hinglish: Mean 2.5 hai, squared deviations ka sum 5, isliye variance 5/4 = 1.25. General formula (n squared - 1)/12 se bhi yahi aata hai.

Q6. How does adding a constant to every observation affect the mean and the variance?

Adding a constant a to every observation adds a to the mean, because the sum of the observations grows by n times a. Each deviation from the new mean is the same as the old deviation, so the squared deviations are unchanged and the variance and standard deviation stay the same. This is why changing the origin of a scale, for example measuring temperatures in a shifted scale, does not change the spread of the data.

Hinglish: Constant add karne se mean utna hi badh jaata hai, lekin deviations same rehte hain, isliye variance aur standard deviation nahi badalte.

Q7. A class of 20 students has mean marks 60 and variance 16. Another class of 30 students has mean marks 70 and variance 25. Find the combined mean.

The combined mean is the weighted average: (20 times 60 + 30 times 70) divided by 50, which is (1200 + 2100)/50 = 3300/50 = 66. The combined mean lies between the two class means and is closer to the mean of the larger class. To find the combined variance we would also need the between-group term, because the two class means differ.

Hinglish: Combined mean weighted average hota hai: (20 x 60 + 30 x 70)/50 = 66. Combined variance ke liye between-group term bhi chahiye hoga.

Q8. When is the mean deviation about the median preferred to the mean deviation about the mean?

The median is less affected by extreme observations than the mean, so when a data set contains outliers the mean deviation about the median is more stable and more representative. The median also has the property that the sum of absolute deviations from it is minimum, which makes it the natural centre for this measure. For symmetric data the two values are close, and the mean deviation about the mean is usually easier to compute.

Hinglish: Extreme values hone par median zyada stable hota hai, isliye median ke about MD behtar hota hai. Median se absolute deviations ka sum minimum hota hai.

Q9. A data set of ten numbers is summarised as mean 20 and standard deviation 3. One entry, 15, is later found to be 25. Work out the corrected mean.

The old sum of the observations is 10 times 20 = 200. The wrong value 15 is replaced by 25, so the sum increases by 10 and the corrected sum is 210. The corrected mean is 210/10 = 21. To correct the standard deviation we would also update the sum of squares by adding 25 squared minus 15 squared, and then recompute the variance from the corrected mean.

Hinglish: Purana sum 200 tha, 15 ki jagah 25 aane se sum 210 ho gaya, isliye naya mean 21. SD correct karne ke liye sum of squares bhi update karna padega.

Q10. Why is the standard deviation preferred over the variance for describing spread?

The variance is the average of squared deviations, so its unit is the square of the unit of the data. A variance of 25 for marks would be in marks squared, which is hard to interpret. The standard deviation is the square root of the variance, so it comes back to the original unit and can be compared directly with the mean and with the observations. Both measures order data sets in the same way, because taking the square root is monotonic, but the standard deviation is easier to communicate.

Hinglish: Variance ka unit data ke unit ka square hota hai, isliye samajhna mushkil hota hai. Standard deviation uska square root hai, isliye original unit me aata hai aur interpret karna aasan hai.

Quiz below: five questions on range, mean deviation, variance, transformations and combined data. Try them before checking the formula sheet.