Podcast Beta
Questions and Answers
What is the cumulative frequency curve also known as?
Headings for different columns in a table are known as:
If the Mean is 40 and the Mode is 42, how is the distribution characterized?
The sum of deviations from the mean equals:
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Which measure of central tendency is influenced by extreme values?
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What is the mode of the word 'STATISTICS'?
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For shoe sizes, which average is most suitable for averaging?
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What is the geometric mean of the numbers 2, 4, and 8?
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What is the correct formula for the expected value of the product of two independent variables X and Y?
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What is the standard deviation of the data set consisting of five identical numbers, 3?
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What does the first moment about the mean equal?
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In a negatively skewed distribution, which equality holds true?
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If the variance of X is 9, what is the variance of the transformation 2X + 4?
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What is the range of the set of scores: 19, 3, 140, 25, 95?
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What is the formula for the Coefficient of Quality Deviation?
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How does the value of standard deviation change?
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What is a quantity calculated from the sample called?
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Grouped data is best classified as which type of data?
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What type of variable is the brand of soap?
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An ogive is best described as what?
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The average of the lower and upper limits of a class is known as what?
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The total angle in a pie diagram is:
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The frequency of a class divided by the total frequency is called what?
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In a percentage frequency distribution, the total of percentage frequencies equals what?
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What is the index number called when all the values are not of equal importance?
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Which index is used to determine the general purchasing power of a country's currency?
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For computing the chain index number, what type of relative is computed?
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What is the index number for the base period always taken as?
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What is the ideal average for computing the chain index number?
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Laspayre's index number is also known as what?
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What type of index number is computed for a single commodity?
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In an index number, what does the price of the base period denote?
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What is the coefficient of skewness for a moderately skewed distribution with a mean price of Rs. 20 and a median price of Rs. 17, given a coefficient of variation of 20%?
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Which of the following is true about the first moments about X = 17.5 from the given frequency distribution?
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What is the quartile deviation for the given observations: 10, 12, 15, 25, 28, 35, 42, 20, 32, 18, 14, 5?
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Which method is ineffective in calculating the mean deviation from the mean of the data set: 2, 5, 6, 6, 8, 9, 12, 13, 16, 23?
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If the first three moments of a distribution about the value 2 are 1, 8, and 20, what does this imply about the variance?
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What does the value index given by the formula Σρη 40 / Σρολο × 100 represent?
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What type of index does CPI fall into?
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Which one of the following formulas was originally designed for calculating a price index number?
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Study Notes
Multiple Choice Questions
- A quantity calculated from the sample is called a statistic.
- Grouped data is secondary data.
- A statistic is a characteristic calculated from sample data.
- The brand of soap is a qualitative variable.
Short Questions
- Continuous variables can take any value within a range, while discrete variables can only take specific, separate values.
- Primary data is collected directly from the source.
- Population refers to the entire group of individuals, items, or data points that are being studied, while sample is a subset of the population.
- Sources of secondary data include government publications, newspapers, journals, and online databases.
- Statistics is the study of collecting, organizing, analyzing, interpreting, and presenting data.
- Population refers to the entire group of individuals, items, or data points that are being studied, while sample is a subset of the population.
- Descriptive statistics involves summarizing and presenting data in a meaningful way, using measures like mean, median, mode, and standard deviation.
- Statistics as a discipline of science is concerned with the collection, analysis, interpretation, and presentation of data to gain insights and make informed decisions.
- Population refers to the entire group of individuals, items, or data points that are being studied.
- Secondary data is data that has already been collected and is available for use.
- Characteristics of statistics include its dependence on data, its quantitative nature, its ability to be summarized and analyzed, and its use in making informed decisions.
- Statistics is the study of collecting, organizing, analyzing, interpreting, and presenting data, while data refers to the raw facts, figures, and information that is collected.
- Primary data is collected directly from the source, while secondary data is data that has already been collected and is available for use.
Chapter 02: Presentation of Data
Multiple Choice Questions
- An ogive is a frequency curve.
- The difference between the largest and smallest value is the range.
- The total angle of a pie diagram is 360°.
- The average of the lower and upper limits of a class is called the class mark.
- A cumulative frequency curve is also called an ogive.
- A pie diagram consists of a circle.
- The frequency of a class divided by the total frequency is called the relative frequency.
- There are two methods used for the collection of data.
- In a percentage frequency distribution, the total of percentage frequencies is 100.
- The graphic representation of the frequency distribution is a histogram.
- A graph of a cumulative frequency curve is called an ogive.
- Headings for different columns in a table are called column captions.
Short Questions
- Classification involves grouping data into categories, while tabulation involves presenting data in a structured table.
- A frequency polygon is constructed by plotting the midpoints of each class interval on the X-axis and the corresponding frequencies on the Y-axis, and then connecting the points with straight lines.
- A histogram is a bar graph where the bars are adjacent to each other, while a frequency polygon is a line graph that connects the midpoints of each class interval.
- The main parts of a table include the title, stub, column captions, body, and prefactory note.
- Simple classification involves grouping data into a single category, while two-way classification involves grouping data into two or more categories.
- Class marks are the midpoints of each class interval.
- Frequency distribution is a table that shows the frequency of occurrence of each class interval.
- One-way classification involves grouping data into a single category, while two-way classification involves grouping data into two or more categories.
- Classification is the process of organizing data into meaningful categories or groups.
- A histogram is a bar graph where the bars are adjacent to each other, and the height of each bar represents the frequency of the corresponding class interval.
- Class boundary is the dividing line between two consecutive class intervals.
- Class limits are the highest and lowest values that can be included in a class interval, while class boundaries are the points that divide the class intervals.
- Relative frequency is the proportion of observations that fall within a particular class interval.
Chapter 03: Measures of Central Tendency
Multiple Choice Questions
- If the mean = 40 and mode = 42, the distribution is negatively skewed.
- The sum of deviations from the mean is zero.
- The modal letter of the word "STATISTICS" is S and T.
- Arithmetic mean is affected by extreme values.
- The observation that occurs the maximum number of times is called the mode.
- The arithmetic mean for X1 and X2 is (X1 + X2) / 2.
- If Σ(x - 20) = 25, Σ(Χ - 15) = 0, and 2(x - 10) = -10, then the arithmetic mean is 15.
- The suitable average for averaging shoe sizes is mode.
- The mean is affected by a change in scale.
- The geometric mean of 2, 4, 8 is 8.
- A symmetrical distribution has mean = 4, then its mode is 4.
- If a and b are two values, then the geometric mean is √(a x b).
Short Questions
-
Weighted arithmetic mean:
- English: (73 * 3) = 219
- Urdu: (85 * 3) = 255
- Maths: (92 * 4) = 368
- Eco: (65 * 3) = 195
- Total Marks = 219 + 255 + 368 + 195 = 1037
- Total Weights = 3 + 3 + 4 + 3 = 13
- Weighted Arithmetic Mean = 1037/13 = 79.77
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Median for the given data:
- Groups: 15-19 20-24 25-29 30-34 35-39
- f: 4 8 12 9 3
- N = 4 + 8 + 12 + 9 + 3 = 36
- N/2 = 36/2 = 18
- Median Class = 25-29
- Lower Limit (L) = 25
- cf of the preceding class = 4 + 8 = 12
- Frequency of the median class (f) = 12
- Class Interval (h) = 4
- Median = L + ( (N/2) - cf ) / f ) * h = 25 + ( (18 - 12) / 12 ) * 4 = 27
-
Arithmetic Mean:
- X: -30
- u: 5
- f: -2 -1 0 1 2 3 5 8 15 20 12 4
- Σfu = (-2 * 5) + (-1 * 8) + (0 * 15) + (1 * 20) + (2 * 12) + (3 * 4) = 45
- Σf = 5 + 8 + 15 + 20 + 12 + 4 = 64
- Arithmetic Mean = X + (Σfu / Σf) * u = -30 + (45 / 64) * 5 = -28.59
-
Harmonic Mean:
- X: 10 12 14 16 18 20
- f: 1 3 7 12 8 1
- Σ (1/X) = (1/10) + (3/12) + (7/14) + (12/16) + (8/18) + (1/20) = 2.86
- Σf = 1 + 3 + 7 + 12 + 8 + 1 = 32
- Harmonic Mean = Σf / Σ (1/X) = 32 / 2.86 = 11.19
-
Harmonic Mean, Mode, and P40:
- Weight: 40-44 45-49 50-54 55-59
- f: 20 30 40 10
- Σf = 20 + 30 + 40 + 10 = 100
- Σ (1/X) = (20/42) + (30/47) + (40/52) + (10/57) = 2.34
- Harmonic Mean = Σf / Σ (1/X) = 100 / 2.34 = 42.74
- Modal Class = 50-54 (Highest frequency)
- L = 50
- f = 40
- fm = 30
- f'm = 10
- h = 5
- Mode = L + ((f - fm) / (2f - fm - f'm)) * h = 50 + ((40 - 30) / (2 * 40 - 30 - 10)) * 5 = 51.67
- P40 = 40th Percentile
- Class Interval = 45-49
- Lower Limit = 45
- cf of the preceding class = 20
- f = 30
- h = 5
- P40 = L + ((40N/100) - cf) / f) * h = 45 + ((40 * 100 / 100) - 20) / 30) * 5 = 46.67
-
Harmonic Mean and Arithmetic Mean:
- Reciprocal: 0.0267, 0.0235, 0.0211, 0.0191, 0.0174, 0.0160, 0.0148
- Harmonic Mean = n / (Σ (1/X)) = 7 / ((1/0.0267) + (1/0.0235) + (1/0.0211) + (1/0.0191) + (1/0.0174) + (1/0.0160) + (1/0.0148)) = 0.0188
- Arithmetic Mean = ΣX / n = (0.0267 + 0.0235 + 0.0211 + 0.0191 + 0.0174 + 0.0160 + 0.0148) / 7 = 0.0198
-
Annual percentage average increase:
- Year 1: 10% increase
- Year 2: 20% increase on year 1 salary
- Year 3: 25% increase on year 2 salary
- Let initial salary = 100
- Year 1 salary = 110
- Year 2 salary = 110 * 1.20 = 132
- Year 3 salary = 132 * 1.25 = 165
- Total increase over 3 years = 165 - 100 = 65
- Average annual increase = 65 / 3 = 21.67%
-
Average of speeds:
- 10 km/h, 20 km/h, 25 km/h
- Average speed = (10 + 20 + 25) / 3 = 18.33 km/h
-
Mode and Median:
- Wages: 4-6 6-8 8-10 10-12 12-14 14-16
- Employees: 13 110 180 105 18 8
- Modal Class = 8-10 (highest frequency)
- L = 8
- f = 180
- fm = 110
- f'm = 105
- h = 2
- Mode = L + ((f - fm) / (2f - fm - f'm)) * h = 8 + ((180 - 110) / (2 * 180 - 110 - 105)) * 2 = 8.78
- N = 13 + 110 + 180 + 105 + 18 + 8 = 434
- N/2 = 434/2 = 217
- Median Class = 8-10
- L = 8
- cf of the preceding class = 13 + 110 = 123
- f = 180
- h = 2
- Median = L + ((N/2) - cf) / f) * h = 8 + ((217 - 123) / 180) * 2 = 8.97
Chapter 04: Measures of Dispersion
Multiple Choice Questions
- If X and Y are two independent variables, then E(XY) = E(X).E(Y).
- The standard deviation of 3, 3, 3, 3, 3 is equal to zero.
- The first moment about the mean is equal to zero.
- In a negatively skewed distribution Mean < Mode.
- If var (X) = 9, then var (2X + 4) is 36.
- The range of the scores 19, 3, 140, 25, 95 is 140.
- The coefficient of Quality Deviation is (Q3 - Q1) / (Q3 + Q1).
- The value of standard deviation changes by a change in scale.
Short Questions
-
The first four moments about X = 17.5:
- Classes: 5-9 10-14 15-19 20-24 25-29
- Frequency: 5 8 12 10 5
- Midpoint (X): 7 12 17 22 27
- (X-17.5): -10.5 -5.5 0 4.5 9.5
- (X-17.5)^2: 110.25 30.25 0 20.25 90.25
- (X-17.5)^3: -1157.625 -166.375 0 410.625 857.1875
- (X-17.5)^4: 12155.0625 1769.53125 0 1850.3125 8146.921875
- Σf = 5 + 8 + 12 + 10 + 5 = 40
- Σf(X-17.5) = (5*(-10.5)) + (8*(-5.5)) + (120) + (104.5) + (5*9.5) = 10
- Σf(X-17.5)^2 = (5110.25) + (830.25) + (120) + (1020.25) + (5*90.25) = 1200
- Σf(X-17.5)^3 = (5*(-1157.625)) + (8*(-166.375)) + (120) + (10410.625) + (5*857.1875) = 2000
- Σf(X-17.5)^4 = (512155.0625) + (81769.53125) + (120) + (101850.3125) + (5*8146.921875) = 130000
- First moment about X = 17.5 = Σf(X-17.5) / Σf = 10/40 = 0.25
- Second moment about X = 17.5 = Σf(X-17.5)^2 / Σf = 1200/40 = 30
- Third moment about X = 17.5 = Σf(X-17.5)^3 / Σf = 2000 / 40 = 50
- Fourth moment about X = 17.5 = Σf(X-17.5)^4 / Σf = 130000 / 40 = 3250
-
Quartile Deviation:
- Observations: 10, 12, 15, 25, 28, 35, 42, 20, 32, 18, 14, 5
- Arrange in ascending order: 5, 10, 12, 14, 15, 18, 20, 25, 28, 32, 35, 42
- N = 12
- Q1 = (N+1)/4 = (12+1)/4 = 3.25th Observation = (12 + 14)/2 = 13
- Q3 = 3(N+1)/4 = 3(12+1)/4 = 9.75th Observation = (28 + 32)/2 = 30
- Quartile Deviation = (Q3 - Q1) / 2 = (30 - 13) / 2 = 8.5
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First four moments about the mean:
- Data: 45, 32, 37, 46, 39, 36, 41, 48 and 36
- ΣX = 350
- Mean = ΣX / n = 350 / 9 = 38.89
- (X-Mean): 6.11 -6.89 -1.89 7.11 0.11 -2.89 2.11 9.11 -2.89
- (X-Mean)^2: 37.33 47.47 3.57 50.57 0.01 8.35 4.45 83.00 8.35
- (X-Mean)^3: 228.30 -327.83 -6.74 359.86 0.01 -24.24 9.37 759.03 -24.24
- (X-Mean)^4: 1395.35 2254.08 12.81 2566.95 0.001 69.87 19.68 6904.18 69.87
- First moment about mean = Σ(X-Mean) / n = 0/9 = 0
- Second moment about mean = Σ(X-Mean)^2 / n = 291.56 / 9 = 32.39
- Third moment about mean = Σ(X-Mean)^3 / n = 759.13 / 9 = 84.35
- Fourth moment about mean = Σ(X-Mean)^4 / n = 10879.14 / 9 = 1208.79
-
Mean Deviation from mean:
- Data: 2, 5, 6, 6, 8, 9, 12, 13, 16, 23
- ΣX = 90
- Mean = ΣX / n = 90 / 10 = 9
- |X-Mean|: 7 4 3 3 1 0 3 4 7 14
- Σ |X-Mean| = 42
- Mean Deviation from mean = Σ |X-Mean| / n = 42 / 10 = 4.2
-
Lower Quartile:
- Deviation: -12, -8.5, 3.0, 0, 2.5, 6.6, 9.2, 1.6, 0.5 and 0.4
- X = Deviation + 22.5
- X: 10.5, 14, 25.5, 22.5, 25, 29.1, 31.7, 24.1, 23, 22.9
- Arrange in ascending order: 10.5, 14, 22.5, 22.9, 23, 24.1, 25, 25.5, 29.1, 31.7
- N = 10
- Q1 = (N+1)/4 = (10+1) / 4 = 2.75th Observation = (22.5 + 22.9) / 2 = 22.7
-
Variance and Skewness:
- Moments about 2: 1, 8 and 20
- Mean = 2 + 1 = 3
- Variance = 8 - 1^2 = 7
- Third moment about the mean = 20 - 3 * 8 + 3 * 1^1 - 1^3 = -1
- Coefficient of Skewness = -1 / 7^(3/2) = -0.054
- The distribution is negatively skewed as the skewness coefficient is negative.
-
Coefficient of Skewness:
- Classes: 40-50 50-60 60-70 70-80 80-90
- F: 4 8 16 8 4
- Σf = 4 + 8 + 16 + 8 + 4 = 40
- N/2 = 40/2 = 20
- Median Class = 60-70
- L = 60
- cf of the preceding class = 4 + 8 = 12
- f = 16
- h = 10
- Median = L + ((N/2) - cf) / f) * h = 60 + ((20 - 12) / 16) * 10 = 65
- Mode Class = 60-70 (highest frequency)
- L = 60
- f = 16
- fm = 8
- f'm = 8
- h = 10
- Mode = L + ((f - fm) / (2f - fm - f'm)) * h = 60 + ((16 - 8) / (2 * 16 - 8 - 8)) * 10 = 65
- Mean = (40 * 45) + (50 * 8) + (60 * 16) + (70 * 8) + (80 * 4) / 40 = 61.25
- Coefficient of Skewness = (Mean - Mode) / Standard Deviation = (61.25 - 65) / 10.21 = -0.36
-
Variance:
- Classes: 10-19 20-29 30-39 40-49 50-59
- F: 5 25 5 40 10
- Midpoint (X): 14.5 24.5 34.5 44.5 54.5
- Σf = 5 + 25 + 5 + 40 + 10 = 85
- ΣfX = (5 * 14.5) + (25 * 24.5) + (5 * 34.5) + (40 * 44.5) + (10 * 54.5) = 3090
- Mean = ΣfX / Σf = 3090 / 85 = 36.35
- (X-Mean): -21.85 -11.85 -1.85 8.15 18.15
- (X-Mean)^2: 477.42 140.42 3.42 66.42 329.42
- Σf(X-Mean)^2 = (5 * 477.42) + (25 * 140.42) + (5 * 3.42) + (40 * 66.42) + (10 * 329.42) = 12860
- Variance = Σf(X-Mean)^2 / Σf = 12860 / 85 = 151.3
-
Coefficient of Skewness:
- Q2 = 26.01, Q3 = 38.29, Q1 = 13.73
- Coefficient of Skewness = (Q3 + Q1 - 2Q2) / (Q3 - Q1) = (38.29 + 13.73 - 2 * 26.01) / (38.29 - 13.73) = 0.29
-
Coefficient of Variation:
- (i) n=150, Σ(Χ - 100) = 180
- (ii) Σ(Χ - 100)² = 245320.
- Mean = (Σ(Χ - 100) / n) + 100 = (180/150) + 100 = 101.2
- Variance = (Σ(Χ - 100)² / n) - ((Σ(Χ - 100) / n))² = (245320 / 150) - ((180/150))² = 1635.44 - 1.44 = 1634
- Standard Deviation = √Variance = √1634 = 40.42
- Coefficient of Variation = (Standard Deviation / Mean) * 100 = (40.42 / 101.2) * 100 = 40%
-
Mean Deviation about Mean:
- Heights: 88.03, 94.50, 94.90, 95.50, 84.60
- ΣX = 457.53
- Mean = ΣX / n = 457.53 / 5 = 91.51
- |X-Mean|: 3.48 2.99 3.39 3.99 6.91
- Σ |X-Mean| = 20.76
- Mean Deviation about Mean = Σ |X-Mean| / n = 20.76 / 5 = 4.15
- Mean Coefficient of Dispersion = (Mean Deviation about Mean / Mean) * 100 = (4.15 / 91.51) * 100 = 4.54%
-
Moments about the mean:
- First three moments about the value 2: 1,16 and -40
- Mean = 2 + 1 = 3
- Variance = 16 - 1^2 = 15
- Third moment about mean = -40 - 3 * 16 + 3 * 1^1 - 1^3 = -86
Chapter 05: Index Numbers
Multiple Choice Questions
- If all the values consider equal importance in the index number, it is called a weighted index
- The index number given by Σρη 40 / Σρολο × 100 is a value index.
- CPI falls in the category of an aggregative index.
- The formula Σρητο x100 / Σροφο is a price index number.
- If all the values are not of equal importance, the index number is called unweighted.
- The general purchasing power of the currency of a country is determined by the wholesale price index.
- For computing the chain index number, we compute link relative.
- The most suitable average for computing chain index number is geometric mean.
- The index number for the base period is always taken as 100.
- Laspeyre's index number is ΣΡπαπ / ΣΡπο.
- An index number computed for a single commodity is called a simple index.
- The price of the base period is denoted by Pon.
- In index numbers, the base year should be a normal year.
- Laspayre's index number is also named as base year weighted.
Short Questions
- An index number is a statistical measure that compares the value of a variable in a given period to its value in a base period.
- Three different types of index numbers are price index, quantity index, and value index.
-
To find Paasche's Index number:
- We need to know Laspeyre's Index number, Paasche's Index number, and Fisher's Index Number using the formula:
- Fisher's Index number = √(Laspeyre's Index number * Paasche's Index number)
- Paasche's Index number = Fisher's Index number² / Laspeyre's Index number
- Paasche's Index number = (120)² / 108 = 133.33
- We need to know Laspeyre's Index number, Paasche's Index number, and Fisher's Index Number using the formula:
- The geometric mean is the most suitable average in the WPI as it gives equal weightage to all price changes.
-
Paasche's Price Index number:
- Epq1 = 850
- Ep1q1 = 1210
- Paasche's Price Index number = (Ep1q1 / Epq1) * 100 = (1210 / 850) * 100 = 142.35
- An unweighted index number is a simple average of the price relatives, where each price relative is given equal weight.
- In the fixed base method, the base year is fixed for all periods. In the chain base method, the base year changes with each period.
- Volume (quantity) index number measures changes in the quantity of goods or services produced or consumed.
-
Link relative:
- It is the ratio of the value of a variable in a given period to its value in the preceding period.
- We calculate it by dividing the value of a variable in the current period by its value in the preceding period and multiplying by 100.
-
Paasche's Index:
- Fisher's Ideal Index = 117.84
- Laspeyre's Index = 117.9
- Paasche's Index = (Fisher's Ideal Index² / Laspeyre's Index) = (117.84)² / 117.9 = 117.78
- Price index number measures changes in the price of goods or services over time.
- Composite index number is an index number that measures changes in the aggregate value of a group of goods or services.
-
Chain base method:
- It is a method of index number calculation where the base year is changed for each period.
- Instead of comparing all periods to a fixed base year, each period is compared to the previous period.
-
Uses of index numbers:
- To measure inflation
- To track economic growth
-
Fisher's Price Index number:
- Epoq = 322
- Ep1q0 = 340
- Ep1q1 = 362
- Epoq1 = 326
- Fisher's Price Index = √( (Ep1q0 / Epoq0) * (Ep1q1 / Epoq1) ) * 100 = √( (340 / 322) * (362 / 326) ) * 100 = 108.78
- Composite index number is an index number that measures changes in the aggregate value of a group of goods or services.
-
Link relatives:
- Link relative is the ratio of the value of a variable in a particular period to its value in the preceding period.
-
Consumer price index (CPI):
- It is a measure of the average change over time in the prices paid by urban consumers for a basket of consumer goods and services.
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