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Question Bank for Statistics Part – 1
Chapter # 1: Introduction to Statistics
1. Define statistics.
Ans: It is a branch of science that deals in collection, processing, presentation, analysis and interpretation of numerical data in order to make decision.
2. Define population and sample.
Ans: Population: The total number of objects having some common characteristics is called as population. e.g. (number of cars, number of trees, number of chairs)
Sample: Any small part of population showing some common characteristics is called as sample.
3. Write down characteristics of statistics.
There are some important characteristics of statistics: Statistics are aggregate of facts. Statistics are numerically expressed. Statistics are collected in a systematic manner. Statistics are collected with a definite purpose.
4. Define data.
Ans: Data are the individual pieces of factual information recorded and used for the purpose of analysis. It is the raw information from which statistics are being created.
5. What is primary data?
Ans: The data which has just been collected from the source and has not gone through any kind of statistical treatment like sorting and tabulation is called as primary data.
Example: The data in the population census reports are primary because these ae collected, compiled and published by the population census organization.
6. Define secondary data.
Ans: The data which has already been collected by someone, that has undergone a statistical treatment like sorting and tabulation is called as secondary data.
Example: The data in economic survey of Pakistan is secondary because these are originally collected by the Federal Bureau of statistics, the State Bank of Pakistan.
7. Sources of Primary data.
Ans: Primary data is collected by the following sources:
i. Direct personal observation. ii. Registration. iii. Investigation through enumerators. iv. Information through mailed questionnaire.
v. Through local correspondents. vi. Through telephone.
8. Sources of secondary data.
Ans: Secondary data is collected by the following sources:
i. Through government organizations. ii. Through semi-government organizations. iii. Through teaching and research organizations. iv. Through newspapers. v. Through internet.
9. Describe any two uses of statistics.
Ans: the following are the uses of statistics:
i. It helps in collection of data. ii. It is used for presentation of data. iii. It helps in processing of data. iv. It is used for comparison of data.
10. Define variable.
Ans: A measureable quantity which varies from one individual or object to another is called variable. For example: Weight, height, time etc.
11. What is constant?
Ans: The value which remains the same from person to person is called as constant. For example, value of pi.
12. How many types of variable?
Ans: There are two types of variables:
1- Quantitative variable 2- Qualitative variable
Quantitative variable: The variables which can be expressed numerically with or without units are known as quantitative variables. For example: Time, Heights, weights, etc.
Qualitative variable: The variable which can be expressed in the form of qualities like: eye color, hair color, IQ level etc.
13. Define discrete variable.
Ans: The variable which can be countable is called discrete variable. For example, number of cars, number of chairs in the classroom, number of houses in the street etc.
14. Define continuous variable.
Ans: A measureable variable is known as continuous variable. for example, height, weight, length etc.
15. Define inferential statistics.
Ans: The phase of statistics that is concerned with the procedures and methodology for obtaining valid conclusions is called inferential statistics.
Chapter # 2: Presentation of data
1. What do you meant by the term ‘classification’?
Ans: The process of arranging data into classes or categories according to some common characteristics present in the data is called classification.
Examples:
1- Sorting of letters in a post office, the letters are classified according to the cities
2- The students of the college are classified according to their hair color.
3- The students of the university are classified according to their heights.
2. Write down the types of classification.
Ans: The data may be classified according to one, two or many characteristics:
i. One-way Classification: When the data is classified by one characteristic, it is called one way classification. ii. Two-way classification: When the data is classified by two characteristics, it is called two-way classification. iii. Many-ways classification: When the data is classified by many characteristics, it is called many-way classification.
3. How many forms of classification?
Ans: There are four main forms of classification:
i. Quantitative Classification. ii. Qualitative Classification. iii. Geographical Classification. iv. Chronological or temporal Classification.
Quantitative Classification: When the data is classified by quantitative characteristics, it is called quantitative classification. For example, weight, height, income, etc. Qualitative Classification: When the data is classified by qualitative characteristics, it is called qualitative classification. For example, sex, religion, color, intelligence, etc. Geographical Classification: When the data is classified by geographical regions or locations, it is called geographical or spatial classification. For example, provinces, divisions, districts, cities, etc.
Chronological Classification: When the data is classified according to its time of occurrence, it is called chronological or temporal classification. For example, years, months, weeks, days, etc.
4. Define tabulation.
Ans: The process of arranging data into rows and columns is called tabulation.
5. Differentiate between class limits and class boundaries.
Ans:
Class Limit : Each class start from a lower limit and ends at an upper limit.
For Example:
Class 20 ---- 30 30 ------ 40 40 -------- 50 50 -------- 60
In the above example 20 is lower limit and 30 is upper limit of the first class and same goes for the others.
Class Boundary: Class limits are called class boundaries if the upper limit of 1st^ class equals to the lower limit of 2nd^ class and so on. Hence if classes progress without break, class limits are called class boundaries. In case we don’t have equal classes then it can be converted easily by increasing the upper class limit and decreasing the lower class limit by the same amount so that there are no gaps left among the classes. For example add 0.5 in upper class and 0.5 in lower class.
Class 20 ----29 30 ------39 40 --------49 50 --------
Class Boundaries 19.5 ------29.5 29.5 -------39.5 39.5 ------- 49.5 49.5 ---------59.
6. What is meant by relative frequency of a class?
Ans: The frequency of a class divided by total frequency of the class is called relative frequency.
7. Define frequency distribution.
Ans: A frequency distribution is a tabular arrangement of data in which various items are arranged into classes and the number of items falling in each class (Called class frequency).
8. Define class mark.
Ans: The class mark or the midpoint is that value which divides a class into two equal parts. It is obtained by adding the lower and upper class limits or class boundaries of a class and dividing the resulting total by 2.
9. Define histogram.
Ans: A histogram consists of a set of adjacent rectangles having class boundaries along the x-axis and frequencies along y-axis.
10. What is the table?
Ans: A systematic arrangement of data into rows and columns is called table.
11. What is grouped data?
Ans: Data presented in the form of a frequency distribution is called grouped data.
12. Write down the main parts of the table.
Ans: Following are the different parts of a table out of which first four are main part:
i. Title ii. Column caption & box head iii. Row caption & stub iv. Body of the table v. Prefatory note
Chapter # 3: Measures of central location
1. What is meant by measures of central tendency?
Ans: The averages tend to lie in the center of a distribution they are called measures of central tendency. They are also called measures of location because they locate the center of a distribution.
2. Write the types of averages.
Ans: The most commonly used averages are:
i. Arithmetic mean ii. Geometric mean iii. Harmonic mean iv. Median v. Mode
3. What are two qualities of a good average?
Ans: Properties of a good average are given below: i. It is well defined ii. It is easy to calculate iii. It is easy to understand iv. It is based on all the values v. It is capable of mathematical treatment
4. Mean of 5 values is 70. Find the sum of values.
Ans: 𝑋 = (^) 𝑛𝑥
𝑥 = 350 Ans.
5. In a moderately skewed distribution, the values of mean and median are 120 and 110 respectively, find the value of mode.
Ans: 𝑀𝑜𝑑𝑒 = 3𝑀𝑒𝑑𝑖𝑎𝑛 − 2 𝑀𝑒𝑎𝑛
𝑀𝑜𝑑𝑒 = 3(110) − 2(120)
𝑀𝑜𝑑𝑒 = 90 Ans.
6. Given u = (x – 170)/5, ∑fu = 100, ∑f = 200 , find arithmetic mean.
Ans: By coding method:
𝑓𝑈 𝑓 ×^ ℎ^ Where;^ 𝑈^ =^
𝑥−𝐴 ℎ
𝑋 = 172.5 Ans.
7. Write down any two mathematical properties of arithmetic mean.
Ans: There are following mathematical properties of Arithmetic Mean:
i. The sum of deviations of all observations from their mean is zero. i.e. 𝑋 − 𝑋 = 0 ii. The sum of squares of deviations of all observations from their mean is minimum. i.e. 𝑋 − 𝑋 2 is minimum
iii. The mean of a constant is constant itself. i.e. If 𝑋 = 𝑎 then 𝑋 = 𝑎 iv. The mean is affected by change of origion and scale. If we add or subtract a constant from all the values or multiply or divide all the values by a constant, the mean is affected by the respective change. i.e. If 𝑌 = 𝑋 ± 𝑎 then 𝑌 = 𝑋 ± 𝑎 , If 𝑌 = 𝑎 ± 𝑏𝑋 then 𝑌 = 𝑎 ± 𝑏𝑋 , If 𝑌 =
𝑋 𝑎 then^ 𝑌^ =^
𝑋 𝑎
8. Define mode and give its formula in case of grouped data.
Ans: The most repeated value in a data is called mode. It is denoted by 𝑋
Formulas;
For ungrouped data: 𝑋 = The most frequent value in a data
For grouped data: 𝑋 = 𝑙 + (^) 𝑓 𝑓𝑚^ −𝑓^1 𝑚 −𝑓 1 +^ 𝑓𝑚 −𝑓 2
× ℎ
9. Find the mode of 3, 3, 7, 8, 10, 11, 10, 12, and 10.
Ans: 10 is most frequent value in the given data and then called the mode.
10. Define the median with formula.
Ans: The value which divides the ordered data into two equal parts is called as median. It is denoted by 𝑋.
11. Write down the properties of median.
Ans:
- a constant “a” is added to each of the n observations y 1 , y 2 , ……. Yn having median M, then the median of y 1 +a, y 2 +a, ……. yn+a would be “a+M”.
- The sum of the absolute deviations of the observations from their median is minimum i.e.,
∑|y – median| is minimum
- For a symmetrical distribution median is equidistant from the first and third quartiles i.e.,
Q 3 – Median = Median = Median – Q 1
12. What are the advantages and disadvantages of median?
Ans:
Merits of Median:
- It is quick to find.
- It is not much affected by exceptionally large or small values in a data.
- It is suitable for skewed distribution.
Demerits of Median:
- It is not rigidly defined.
- It is not readily suitable for algebraic development.
- It is less stable in repeated sampling experiments than the mean.
- It is not based on all the observations.
Total 0.
H.M. = 8.571 Ans.
18. Define weighted mean.
Ans: When all the values in the data are not of equal importance then we assign them certain numerical values to express their relative importance. These assigned values are called weights. The average of these weights with values is called weighted mean. The weights may be the quantities consumed or the numerical coefficient and are generally
denoted by (^) . The weighted mean denoted by “ (^) yw ” of a set of ‘n’ values say
y 1 , y 2 ,..... yn With weights 1 , 2 ,..... n is then given by:
n
n n
w w w w
wy wy wy
y
1 2
1 1 1 1
i
i i
w w
wy
y
Where; i = 1, 2, 3, 4 …n
19. Define harmonic mean.
Ans: Harmonic mean is defined as the reciprocal of the mean of the reciprocals of the items in a series. It is the ratio of the number of items and the sum of reciprocal of items.
H =
x
f
f
20. Calculate geometric mean of X = 1, 1, 27.
Ans: by definition;
𝐺. 𝑀. = 𝑛𝑥1.𝑥2.𝑥3. …. 𝑥𝑛
𝐺. 𝑀. = 3 1.1.
𝐺. 𝑀. = 3 Ans.
21. Write down properties of geometric mean.
Ans:
- G.M is always less then A.M. i.e GM < A.M.
- Geometric mean of constant variable is always constant. 22. Write down merits of geometric mean.
Ans:
- It is rigidly defined by a mathematical formula.
- It is based on all values.
- It is less affected by extremely large values.
23. Write down demerits of geometric mean.
Ans:
- It is not calculated if any of the observations is zero or negative.
- In case of negative values, it cannot be computed at all.
- It is not easy to understand. 24. Write down the merits of A.M.
Ans:
- It is rigidly defined by mathematical formula.
- It is easy to calculate.
- It is easy to understand.
- It is based upon all the values.
- It is stable statistics in repeated sampling experiments. 25. Write demerits of A.M.
Ans:
- It is greatly affected by extreme value.
- It cannot be calculated for open-end classes without assuming open ends.
- It gives fallacious and misleading conclusions when there is too much variation in data. 26. Find A.M. if 𝒇𝒙 = 𝟓𝟎𝟎 and 𝒇 = 𝟓𝟎.
Ans: By definition;
𝑋 = 10 Ans.
27. If G.M. of two values is 3. Find the product of two values.
Ans: by definition;
𝐺. 𝑀. = 𝑛𝑥1.𝑥2.𝑥3. …. 𝑥𝑛
2
28. Illustrate the graphically positions of mean, median and mode for frequency curve which are skewed to the right and left.
Ans.
a) For moderately positively skewed distributions, the following empirical relation holds. Mean > Median > Mode
32. If 𝒚𝟏 = 𝟑 with 𝒏𝟏= 𝟑 and 𝒚𝟐 = 𝟒 with 𝒏𝟐=𝟐 , then find 𝒚𝒄.
Ans: By definition;
𝑦𝑐 = 17/5 Ans.
33. Define average.
Ans: An Average is a single value which represents all values of data in some definite way.
For Example: The average income of middle class families is Rs.17000/per month.
34. For a certain frequency distribution, the mean was 40.5 and median was 36. Find mode by using of empirical relation.
Ans: Mode = 3Median – 2 mean
Mode = 3(36) – 2(40.5)
Mode = 27 Ans.
Chapter # 4: Measures of Dispersion
1. Write any two advantages of the range.
Ans:
- It is easy to calculate.
- It is useful measure in small samples. 2. What are the demerits of range?
Ans:
- It is not based on all observations.
- It depends only upon the extremes observations. 3. Define relative dispersion.
Ans: Relative measures of dispersion are calculated as ratios or percentages; for example, one relative measure of dispersion is the ratio of the standard deviation to the mean. Relative measures of dispersion are always dimensionless, and they are particularly useful for making comparisons between separate data sets or different experiments that might use different units. They are sometimes called coefficients of dispersion.
4. What is quartile deviation?
Ans: Half of the difference between the upper and the lower quartiles is called as quartile deviation or semi-inter quartile range. It is denoted by ‘Q.D’
Formula: 𝑄. 𝐷. =
𝑄 3 −𝑄 1 2
5. Write down the merits of quartile deviation.
Ans:
- It is easy to calculate.
- It is not affected by extreme values. 6. Write down the demerits of quartile deviation.
Ans:
- If is not based on all the observations.
- Q.D. will be the same values for all the distribution having the same quartiles. 7. Define mean deviation
Ans: It is defined as the mean of the absolute deviation of observations from mean, median or mode. By absolute deviations we mean that we consider all the deviations as positive. It is denoted by M.D. and calculated as:
|𝑌−𝑀| 𝑛 (Here, M is mean/median/mode)
14. What is the use of coefficient of variation?
Ans: Co-efficient of variation is a relative measure of dispersion and independent of units of measurement and expressed in percentage. It is used to compare the variability of different sets of data. The group which has lower value of coefficient, coefficient of variation is comparatively more consistent.
15. What do you say about the relative dispersion of 5, 5, 5 and 5?
Ans: Relative dispersion would be zero, because the absolute dispersion of constant is zero.
16. If S^2 = 36 and X = 18, what is coefficient of variation?
Ans: By definition:
× 100
Solution;
S^2 = 36 ; S = 6 and X = 18 put in above equation
𝑪. 𝑽. = 𝟑𝟑. 𝟑𝟑𝟑
17. If variance of the value of ‘ X ’ is 25, what is the standard deviation of X****?
Ans: By taking square root of variance of x we get standard deviation of x. S.D.(x) = 5.
18. If S.D(X) = 10, then find the standard deviation of 5X?
Ans: By property of Standard deviation: 𝑆. 𝐷. 𝑎𝑥 = 𝑎 𝑆. 𝐷. 𝑥
𝑆. 𝐷. 5 𝑥 = 5 𝑆. 𝐷. 𝑥
𝑆. 𝐷. 5 𝑥 = 5 10
𝑺. 𝑫. 𝟓𝒙 = 𝟓𝟎
19. What is meant by symmetry?
Ans: In a symmetrical distribution, a deviation below the mean is equal to the corresponding deviation above the mean. This is called symmetry.
20. Define skewness.
Ans: Skewness is the lack of symmetry in a distribution around central (men, median or mode).
21. What are the types of measures of dispersion?
Ans: There are two types of measures of dispersion:
Absolute Measure Relative Measure
22. Define the term variance.
Ans: Mean of squares of deviation of all the observations from their mean is called as variance. It is denoted by S^2.
𝑆^2 =
(𝑥−𝑥 )^2 𝑛
OR
𝑆^2 =
𝑥^2
2
23. For a symmetrical distribution S.D. = 2. What is value of 4th^ moment about mean for mesokurtic data?
Ans: The dimensionless measure of kurtosis based on the moments is𝛽 2 = 𝜇 𝜇^4 22
. If 𝛽 2 = 3, the distribution is mesokurtic
(normal).
Variance = 4 by using S.D. = 2;
Forth moment about mean = 𝜇 4 = 48 Ans.
24. What do you know about kurtosis?
Ans: The word kurtosis is used to indicate the length of the tails and peakedness of symmetrical distributions. Symmetrical distribution may be platykurtic (more peaked), mesokurtic (normal) or leptokurtic (bit flat).
Lepto-kurtic
Meso-kurtic
Platy-kurtic
25. If b 2 = 3 and m 4 = 1875, then what will be the standard deviation?
Ans: By using formula;
𝛽 2 =
𝜇 4 𝜇 22
𝜇 2 = 25 (Variance)
S.D. = 5 Ans.
26. What is meant by absolute dispersion?
Ans: It can be defined as such a way that they have units (meters, grams) same as those of original measurements. There are following measures of absolute dispersion:
- Range
- Quartile Deviation
- Mean Deviation
- Variance
- Standard Deviation 27. Write four measures of relative dispersion.
Ans: The relative measures of dispersion are given below:
- Co-efficient of Range = 𝑋 𝑋𝑚𝑎𝑥^ −𝑋𝑚𝑖𝑛 𝑚𝑎𝑥 +𝑋𝑚𝑖𝑛
32. Define standard deviation.
Ans: The standard deviation is defined as the positive square root of the mean of the squares of the deviations of values from their mean. In other words, standard deviation is a positive square root of variance. It is denoted by ‘S’.
33. What is meant by dispersion?
Ans: The degree to which numerical data tend to spread about an average value of the data is called as dispersion.
34. Define measure of dispersion.
Ans: A numerical quantity called measure of dispersion that describes the spread of the values in a set of data.
35. Define range.
Ans: The difference between the largest and the smallest observation is called as range. It is denoted by ‘R’.
R = Xmax – Xmin
36. What is the range of Bowley’s coefficient of skewness?
Ans: It lies between -1 to +1.
37. If 𝒖𝟐 = 𝟒 and 𝒖𝟒 = 𝟓𝟔 , find 𝜷𝟐.
Ans: By using given formula;
38. If var(X) = 4, then find var(3X).
Ans: By using of variance property:
𝑣𝑎𝑟(3𝑥) = 9𝑣𝑎𝑟(𝑥)
𝑣𝑎𝑟(𝑦) = 9(4)
𝒗𝒂𝒓(𝒚) = 𝟑𝟔
39. What are the measures of absolute dispersion?
Ans: Some absolute measures of dispersion are:
- Range
- Quartile Deviation
- Mean Deviation
- Variance
- Standard Deviation
40. If var(x) = 10 and y = 5x + 20, then find var(y).
Ans: By property of S.D. ;
𝑦 = 5𝑥 + 20
𝑣𝑎𝑟(𝑦) = 𝑣𝑎𝑟(5𝑥 + 20)
𝑣𝑎𝑟(𝑦) = 25𝑣𝑎𝑟(𝑥)
𝑣𝑎𝑟 𝑦 = 25 10
𝒗𝒂𝒓 𝒚 = 𝟐𝟓𝟎
41. S.D. of a distribution is 4. Find second moment about mean.
Ans: As we know that second moment a bout mean is equals to variance then; variance = 16 (if S.D. = 4)
