Quantitative Reasoning Statistical Basics

Measures of Central Tendency

Measures of central tendency are the specific statistical values which describe the typical values or central position of a dataset. The three most common single statistical values include Mean, Median and Mode.

Arithmetic Mean: Average of all values (sum divided by count).

Geometric Mean: nth root of the product of values (used for growth rates).

Harmonic Mean: The Reciprocal of the average of reciprocals (used for rates like speed).

Median: Middle value when data is ordered.

Quartiles: Values dividing data into four equal parts (Q1=25th, Q2=50th=median, Q3=75th percentile).

Mode: Most frequent value.

Measures of Dispersion

Range: Difference between maximum and minimum values.

Quartile Deviation: Half the interquartile range (Q3 - Q1)/2.

Mean Deviation: Average absolute deviation from the mean.

Variance: Average squared deviation from the mean.

Standard Deviation: Square root of variance (spread in original units).

Coefficient of Variation: (Standard Deviation / Mean) × 100% (relative variability).

Data Distribution Shapes

Symmetric: Balanced, mean = median = mode (e.g., normal distribution).

Skewed Right: Longer right tail, mean > median > mode.

Skewed Left: Longer left tail, mean < median < mode.

Uniform: All values equally likely, flat shape.

Bimodal: Two distinct peaks.

Skewness and Kurtosis

Skewness: Measures asymmetry (positive = right skew, negative = left skew, zero = symmetric).

Kurtosis: Measures tail heaviness (high = heavy tails/peak, low = flat/light tails, zero = normal-like).

Multiple Choice Questions (MCQs)

1. What is the arithmetic mean of the ungrouped data set: 5, 7, 3, 9, 6?

A) 5 B) 6 C) 7 D) 8

2. For grouped data, the arithmetic mean is calculated using:

A) Sum of values divided by number of values

B) Frequencies times midpoints summed and divided by total frequency

C) Product of midpoints raised to 1/n

D) Reciprocal of the sum of reciprocals

3. The geometric mean is most appropriate for which type of data?

A) Rates of growth or ratios

B) Speeds or rates

C) Ordered data for the middle value

D) Most frequent values

4. Calculate the geometric mean for ungrouped data: 4, 9, 16.

A) 9

B) 8

C) 7.5

D) 6.24

5. In grouped data, the geometric mean is used:

A) Logarithms of midpoints weighted by frequencies

B) Sum of absolute deviations

C) Cumulative frequencies

D) Squared deviations

6. The harmonic mean for ungrouped data: 2, 3, 6 is

A) 3 B) 3.27 C) 4 D) 11

7. Harmonic mean is best for:

A) Averages of rates like speed

B) Multiplicative data

C) Finding the middle value

D) Most common value

8. For the ungrouped data: 1, 2, 3, 4, 5, the median is:

A) 2 B) 3 C) 4 D) 5

9. In grouped data, the median is found using:

A) The class with the highest frequency

B) Interpolation in the median class based on cumulative frequency

C) Sum of midpoints

D) Product of values

10. The first quartile (Q1) divides the data such that what percentage is below it?

A) 25% B) 50% C) 75% D) 100%

11. For ungrouped data: 10, 20, 30, 40, 50, Q3 is:

A) 20 B) 30 C) 40 D) 50

12. The mode in ungrouped data is:

A) The average value

B) The middle value

C) The most frequent value

D) The spread value

13. In grouped data, the mode uses the modal class and:

A) Adjacent frequencies in a formula

B) Cumulative sums

C) Logarithms

D) Squared differences

14. The range for ungrouped data: 15, 22, 8, 30, 12 is:

A) 15 B) 22 C) 30 D) 22

15. Quartile deviation is:

A) (Q3 - Q1)/2 B) Max - Min

C) Average absolute deviation

D) Square root of variance

16. Mean deviation measures:

A) Squared deviations from the mean

B) Absolute deviations from the mean

C) Relative variability

D) Tail heaviness

17. Variance is the:

A) Square root of standard deviation

B) Average squared deviation from the mean

C) Half of the interquartile range

D) Most frequent deviation

18. Standard deviation is useful because it is in:

A) Squared units

B) The same units as the data

C) Percentages

D) Logarithmic scale

19. Coefficient of variation compares:

A) Absolute spread

B) Relative variability across datasets

C) Asymmetry

D) Peakedness

20. Positive skewness indicates:

A) Symmetric distribution

B) Longer left tail

C) Longer right tail

D) Flat distribution

Here are the answers to the 20 MCQs:

1. What is the arithmetic mean of the ungrouped data set: 5, 7, 3, 9, 6?

B) 6 (Calculation: (5+7+3+9+6)/5 = 30/5 = 6)

2. For grouped data, the arithmetic mean is calculated using:

B) Frequencies times midpoints summed and divided by total frequency

3. The geometric mean is most appropriate for which type of data?

A) Rates of growth or ratios

4. Calculate the geometric mean for ungrouped data: 4, 9, 16.

B) 8 (Calculation: (4 × 9 × 16)^(1/3) = (576)^(1/3) ≈ 8)

5. In grouped data, the geometric mean is used:

A) Logarithms of midpoints weighted by frequencies

6. The harmonic mean for ungrouped data: 2, 3, 6 is:

B) 3.27 (Calculation: 3/(1/2 + 1/3 + 1/6) = 3/(0.5 + 0.333 + 0.167) ≈ 3/0.917 ≈ 3.27)

7. Harmonic mean is best for:

A) Averages of rates like speed

8. For the ungrouped data: 1, 2, 3, 4, 5, the median is:

B) 3 (Middle value in ordered data)

9. In grouped data, the median is found using:

B) Interpolation in the median class based on cumulative frequency

10. The first quartile (Q1) divides the data such that what percentage is below it?

A) 25%

11. For ungrouped data: 10, 20, 30, 40, 50, Q3 is:

C) 40 (Q3 position: 3(5+1)/4 = 4.5, interpolate: 40)

12. The mode in ungrouped data is:

C) The most frequent value

13. In grouped data, the mode uses the modal class and:

A) Adjacent frequencies in a formula

14. The range for ungrouped data: 15, 22, 8, 30, 12 is:

B) 22 (Calculation: 30 - 8 = 22)

15. Quartile deviation is:

A) (Q3 - Q1)/2

16. Mean deviation measures:

B) Absolute deviations from the mean

17. Variance is the:

B) Average squared deviation from the mean

18. Standard deviation is useful because it is in:

B) The same units as the data

19. Coefficient of variation compares:

B) Relative variability across datasets

20. Positive skewness indicates:

C) Longer right tail

✍ By: Raja Bahar Khan Soomro

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