Statistics · Class 10 Mathematics: 66 flashcards
All 66 flashcards from this kit, free and with no sign-up.
1. Why must the mean of grouped data be estimated?
Show the answer
Grouping discards the individual values, leaving only how many fell in each class.
2. What is a class mark?
Show the answer
The midpoint of a class: half the sum of its lower and upper limits.
3. Find the class mark of the class 20–30.
Show the answer
$\frac{20+30}{2} = 25$.
4. State the direct method formula for the mean.
Show the answer
$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
5. What is the divisor in every mean formula here?
Show the answer
$\sum f_i$, the total frequency — never the number of classes.
6. State the assumed-mean formula.
Show the answer
$\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$, with $d_i = x_i - a$.
7. Why can $a$ be any value at all?
Show the answer
Because the $a$ subtracted in each $d_i$ is added back by the leading $a$; the algebra cancels it exactly.
8. Why is a middle class mark usually chosen for $a$?
Show the answer
It keeps the deviations small and lets positive and negative ones cancel.
9. State the step-deviation formula.
Show the answer
$\bar{x} = a + h \times \frac{\sum f_i u_i}{\sum f_i}$, with $u_i = \frac{x_i - a}{h}$.
10. When can step deviation be used?
Show the answer
Only when all the classes have the same width $h$.
11. What does $u_i$ actually measure?
Show the answer
How many class-widths the class mark lies away from $a$.
12. What is the commonest error in the step-deviation method?
Show the answer
Forgetting to multiply by $h$, which makes the answer far too small.
13. Do the three methods give different answers?
Show the answer
No — all three give exactly the same mean; only the arithmetic differs.
14. For 0–10, 10–20, 20–30, 30–40, 40–50 with frequencies 2, 3, 9, 10, 6, find $\sum f_i x_i$.
Show the answer
$10 + 45 + 225 + 350 + 270 = 900$.
15. Hence find the mean of that distribution.
Show the answer
$\frac{900}{30} = 30$.
16. For that distribution with $a = 25$, what are the $u_i$?
Show the answer
$-2, -1, 0, 1, 2$.
17. And what is $\sum f_i u_i$?
Show the answer
$-4 - 3 + 0 + 10 + 12 = 15$.
18. What happens if you take $d_i = a - x_i$?
Show the answer
Every sign reverses, and the mean comes out on the wrong side of $a$.
19. Which method suits classes of unequal width?
Show the answer
The assumed-mean method; step deviation needs equal widths.
20. What must you show if a question says 'by the step-deviation method'?
Show the answer
The $u_i$ column and the multiplication by $h$; the direct method would lose marks.
21. What is $\sum f_i$ also called?
Show the answer
$n$, the total number of observations.
22. If $n$ is given and one frequency is unknown, what do you have?
Show the answer
An equation: the frequencies must add to $n$.
23. What is the mode of a data set?
Show the answer
The value that occurs most often.
24. What is the modal class?
Show the answer
The class with the largest frequency.
25. Is the modal class the same as the mode?
Show the answer
No — the modal class is an interval; the mode is a single value inside it.
26. State the formula for the mode of grouped data.
Show the answer
$l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)h$.
27. What is $l$ in that formula?
Show the answer
The lower boundary of the modal class — not its class mark.
28. What are $f_0$ and $f_2$?
Show the answer
The frequencies of the classes immediately before and after the modal class.
29. What is the denominator of the fraction?
Show the answer
$2f_1 - f_0 - f_2$, with the modal frequency counted twice.
30. For classes 0–10 … 40–50 with frequencies 2, 3, 9, 10, 6, name the modal class.
Show the answer
30–40, with frequency 10.
31. For that distribution, write down $l$, $f_1$, $f_0$, $f_2$ and $h$.
Show the answer
$l = 30$, $f_1 = 10$, $f_0 = 9$, $f_2 = 6$, $h = 10$.
32. Hence find the mode.
Show the answer
$30 + \frac{1}{5}\times 10 = 32$.
33. Why does the mode always lie inside the modal class?
Show the answer
The fraction $\frac{f_1-f_0}{2f_1-f_0-f_2}$ is between 0 and 1, so the added amount is between 0 and $h$.
34. What quick check should you run on a computed mode?
Show the answer
That it lies between the two boundaries of the modal class.
35. Which way does the mode lean if the class before is nearly as tall?
Show the answer
Towards the lower end of the modal class.
36. Where does the mode sit if both neighbours are equally tall?
Show the answer
Exactly at the midpoint of the modal class.
37. What is $f_0$ when the modal class is the first class?
Show the answer
Zero — there is no class before it.
38. For 0–10, 10–20, 20–30 with frequencies 12, 8, 5, find the mode.
Show the answer
$0 + \frac{12}{24 - 0 - 8}\times 10 = \frac{12}{16}\times 10 = 7.5$.
39. What is a bimodal distribution?
Show the answer
One in which two classes tie for the largest frequency.
40. Why does the syllabus avoid bimodal distributions?
Show the answer
There is no single modal class, so the formula has no unique answer.
41. What must be done with the classes 0–9, 10–19 before using the mode formula?
Show the answer
Make them continuous: they become $-0.5$–$9.5$ and $9.5$–$19.5$.
42. Why does that adjustment matter more for the mode than for the mean?
Show the answer
The mode formula uses the lower boundary and the width directly, while the class mark used by the mean is unaffected.
43. Which average can be used on non-numerical data?
Show the answer
The mode — it needs only counting.
44. Can a data set have more than one mode?
Show the answer
Yes — unlike the mean and median, the mode need not be unique.
45. What is the median?
Show the answer
The middle observation when the data are arranged in order.
46. Why is the median preferred to the mean for incomes?
Show the answer
A few very large values move the mean a long way but leave the median where it is.
47. What is cumulative frequency?
Show the answer
The running total of frequencies up to and including each class.
48. What should the last cumulative frequency equal?
Show the answer
$n$, the total frequency — a free check on your addition.
49. How is the median class found?
Show the answer
It is the first class whose cumulative frequency reaches $\frac{n}{2}$.
50. Is the test $\frac{n}{2}$ or $\frac{n+1}{2}$ for grouped data?
Show the answer
$\frac{n}{2}$, whether $n$ is odd or even.
51. State the median formula for grouped data.
Show the answer
$l + \left(\frac{\frac{n}{2} - cf}{f}\right)h$.
52. What is $cf$ in that formula?
Show the answer
The cumulative frequency of the class BEFORE the median class.
53. What is $f$ in that formula?
Show the answer
The median class's own frequency — not its cumulative frequency.
54. What is $l$?
Show the answer
The lower boundary of the median class.
55. For frequencies 2, 3, 9, 10, 6, write the cumulative frequencies.
Show the answer
2, 5, 14, 24, 30.
56. With $n = 30$, which class is the median class there?
Show the answer
30–40, since 24 is the first cumulative frequency to reach 15.
57. Find the median of that distribution.
Show the answer
$30 + \frac{15 - 14}{10}\times 10 = 31$.
58. Why does the median always lie inside the median class?
Show the answer
By definition $cf < \frac{n}{2} \le cf + f$, so the fraction is between 0 and 1.
59. What is $cf$ when the median class is the first class?
Show the answer
Zero — nothing was counted before it.
60. State the empirical relationship between the three averages.
Show the answer
$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$.
61. Is that relation exact?
Show the answer
No — it is an approximation found by observing many real distributions.
62. A distribution has mean 30 and median 31. Estimate the mode.
Show the answer
$3(31) - 2(30) = 33$.
63. Rearrange the relation to give the median.
Show the answer
$\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3}$.
64. Rearrange it to give the mean.
Show the answer
$\text{Mean} = \frac{3\,\text{Median} - \text{Mode}}{2}$.
65. When is the empirical relation exact?
Show the answer
When the distribution is symmetric, so that all three averages coincide.
66. Is the cumulative frequency GRAPH part of the syllabus?
Show the answer
No — the ogive was removed; cumulative frequency is used only as a column.
Your notes can be a kit like these
Every kit here was made from real study material. Point Veda at a chapter, a PDF, a photo or a link and get notes, flashcards, practice MCQs, games and a scored test — free, in 42 languages.
- Chapter notes
- Flashcards
- Practice MCQs
- Scored tests
- Exam prep
Get Veda free See what students get
Free to start · No ads · Works in 42 languages