Question 1

Choose the correct answer from the four alternatives given below:

(i) The measure of central tendency that does not get affected by extreme values:

(a) Mean

(b) Mean and Mode

(c) Mode

(d) Median

(ii) The measure of central tendency always coinciding with the hump of any distribution is:

(a) Median

(b) Median and Mode

(c) Mean

(d) Mode

Accepted Answer

(i) The measure of central tendency that does not get affected by extreme values is the Median (d). Mean is affected by extreme values, Mode is not necessarily unaffected. Median is resistant to extreme values.

(ii) The measure of central tendency always coinciding with the hump of any distribution is the Mode (d). The Mode is the value that appears most frequently and corresponds to the peak (hump) of the distribution. — Step-by-step:
(i) Mean is the arithmetic average and is sensitive to extreme values. Median is the middle value and is not affected by extremes. Mode is the most frequent value but may or may not be affected by extremes depending on data.
(ii) The Mode corresponds to the highest frequency and thus the peak (hump) of the distribution curve.

Question 2

Answer the following questions in about 30 words:

(i) Define the mean.

(ii) What are the advantages of using mode?

Accepted Answer

(i) Mean is the sum of all observations divided by the number of observations. It represents the average value of the data set.

(ii) Advantages of mode:
- It is easy to understand and calculate.
- It can be used for qualitative data.
- It represents the most common value in the data set.
- It is not affected by extreme values. — Step-by-step:
(i) Calculate mean by adding all data values and dividing by total number.
(ii) Mode advantages include simplicity, applicability to categorical data, and robustness to outliers.

Question 3

Answer the following questions in about 125 words:

(i) Explain relative positions of mean, median and mode in a normal distribution and skewed distribution with the help of diagrams.

(ii) Comment on the applicability of mean, median and mode (hint: from their merits and demerits).

Accepted Answer

(i) In a normal distribution, mean = median = mode, all three measures coincide at the center of the symmetric bell-shaped curve.

In a positively skewed distribution, mode < median < mean. The tail is stretched to the right, so mean is pulled towards higher values.

In a negatively skewed distribution, mean < median < mode. The tail is stretched to the left, so mean is pulled towards lower values.

Diagrams typically show the bell curve for normal distribution and skewed curves with shifted peaks.

(ii) Applicability:
- Mean is useful for quantitative data and is sensitive to extreme values.
- Median is useful when data has outliers or skewed distribution as it is resistant to extremes.
- Mode is useful for qualitative data and to identify the most frequent value but may not represent central tendency well in some cases. — Step-by-step:
(i) Understand distribution shapes and relative positions of measures.
(ii) Evaluate pros and cons of each measure for different data types and distributions.

Question 4

Activity

1. Take an imaginary example applicable to geographical analysis and explain direct and indirect methods of calculating mean from ungrouped data.

Accepted Answer

Direct Method:
Suppose the data set of ungrouped data is: 5, 7, 9, 10, 12.
Mean = (5 + 7 + 9 + 10 + 12) / 5 = 43 / 5 = 8.6

Indirect Method:
Choose an assumed mean (A), say 9.
Calculate deviations d = x - A: (-4, -2, 0, 1, 3)
Sum of deviations Σd = (-4) + (-2) + 0 + 1 + 3 = -2
Mean = A + (Σd / n) = 9 + (-2/5) = 9 - 0.4 = 8.6

Both methods yield the same mean value. — Step-by-step:
Direct method sums all data and divides by count.
Indirect method uses an assumed mean to simplify calculation of deviations and then adjusts the assumed mean accordingly.

Question 5

Which of the following is NOT a measure of central tendency used in geographical data analysis?

Accepted Answer

Variance — Measures of central tendency include mean, median, and mode, which represent a central value of data. Variance is a measure of dispersion, not central tendency.

Question 6

Assertion (A): Measures of central tendency provide a single representative value that best summarizes a set of observations.
Reason (R): Measures of central tendency always lie at the extreme ends of a data distribution.

Accepted Answer

C — Assertion is true because measures of central tendency provide a single representative value summarizing data. Reason is false because these measures usually lie near the center, not at the extremes of the distribution.

Question 7

What is the formula to calculate the mean using the direct method for ungrouped data?

Accepted Answer

\bar{X} = \frac{\sum x}{N} / Mean = sum of observations divided by number of observations — The mean is calculated by adding all values (Σx) and dividing by the total number of observations (N). This formula is fundamental for the direct method with ungrouped data.

Question 8

Calculate the mean rainfall for Malwa Plateau from the following rainfall data (in mm): Indore - 979, Dewas - 1083, Dhar - 833, Ratlam - 896, Ujjain - 891, Mandsaur - 825, Shajapur - 977.

Accepted Answer

926.29 mm — Given: Rainfall values: 979, 1083, 833, 896, 891, 825, 977 mm
Find: Mean rainfall
Formula: \bar{X} = \frac{\sum x}{N}
Solution: Step 1: Sum all rainfall values: 979 + 1083 + 833 + 896 + 891 + 825 + 977 = 6484 mm
Step 2: Number of observations, N = 7
Step 3: Mean = 6484 / 7 = 926.29 mm
Answer: 926.29 mm
Note: Ensure all values are added correctly before division.

Question 9

In the indirect method of calculating mean from ungrouped data, what does the term 'assumed mean' refer to?

Accepted Answer

Assumed mean is a selected value near the center of the data range from which deviations are calculated. For example, in rainfall data between 800 and 1100 mm, 800 mm was chosen as the assumed mean to simplify calculations. — Assumed mean is a constant value chosen to reduce computational complexity by coding data as deviations from this value, facilitating easier calculation of mean in large data sets.

Question 10

Explain the main advantage of using the indirect method over the direct method when calculating mean from ungrouped data.

Accepted Answer

The indirect method simplifies calculations by reducing large raw data values to smaller deviations from an assumed mean, making arithmetic easier and less error-prone. For example, subtracting 800 from rainfall values between 800 and 1100 mm reduces numbers for easier addition. — Indirect method reduces computational effort by coding data as deviations from an assumed mean, which are smaller numbers, facilitating faster and less error-prone calculations especially for large data sets.

Question 11

Calculate the mean wage rate of factory workers using the direct method from the following grouped data:
Classes (Rs./day): 50-70, 70-90, 90-110, 110-130, 130-150
Number of workers: 10, 20, 25, 35, 9

Accepted Answer

102.6 Rs./day — Given: Classes and frequencies
Midpoints (x): 60, 80, 100, 120, 140
Frequencies (f): 10, 20, 25, 35, 9
Find: Mean wage rate
Formula: \bar{X} = \frac{\sum f x}{N}
Solution: Step 1: Calculate fx: (10×60)=600, (20×80)=1600, (25×100)=2500, (35×120)=4200, (9×140)=1260
Step 2: Sum fx = 600 + 1600 + 2500 + 4200 + 1260 = 10,160
Step 3: Sum frequencies N = 10 + 20 + 25 + 35 + 9 = 99
Step 4: Mean = 10,160 / 99 = 102.6 Rs./day
Answer: 102.6 Rs./day

Question 12

Using the indirect method, calculate the mean wage rate from the data in the previous question, assuming the midpoint of the 90-110 class (100) as the assumed mean.

Accepted Answer

102.6 Rs./day — Given: Assumed mean A = 100, Frequencies and midpoints as before
Deviations (d) = x - A: 60-100=-40, 80-100=-20, 100-100=0, 120-100=20, 140-100=40
Calculate fd: f×d = (10×-40)=-400, (20×-20)=-400, (25×0)=0, (35×20)=700, (9×40)=360
Sum fd = -400 - 400 + 0 + 700 + 360 = 260
Sum frequencies N = 99
Formula: \bar{x} = A + \frac{\sum f d}{N}
Solution: Mean = 100 + (260/99) = 100 + 2.63 = 102.63 Rs./day
Answer: 102.6 Rs./day (rounded)

Question 13

Define median as a measure of central tendency.

Accepted Answer

Median is the middle value in an ordered data set that divides the distribution into two equal halves, with an equal number of observations on each side. For example, in a set of mountain peak heights arranged in ascending order, the median is the height at the central position. — Median is a positional average that depends on the order of data, not the actual values, making it less sensitive to extreme values.

Question 14

Calculate the median of the following ungrouped data representing mountain peak heights (in meters): 8126, 8611, 7817, 8172, 8076, 8848, 8598.

Accepted Answer

8172 m — Given: Data values
Step 1: Arrange in ascending order: 7817, 8076, 8126, 8172, 8598, 8611, 8848
Step 2: Number of observations, N = 7
Step 3: Median position = (N+1)/2 = (7+1)/2 = 4th item
Step 4: The 4th value in ordered data is 8172 m
Answer: 8172 m

Question 15

What is the formula to compute the median for grouped data?

Accepted Answer

M = l + \frac{i}{f} \left(\frac{N}{2} - c\right) — The formula for median in grouped data uses the lower limit (l) of the median class, class interval (i), frequency (f) of median class, total observations (N), and cumulative frequency (c) of the class before median class.

them comprehensible. It facilitates data processing. A number of statistical

them comprehensible. It facilitates data processing. A number of statistical — Study Notes

Data Processing

Measures of Central Tendency

Mean

Practice Questions — them comprehensible. It facilitates data processing. A number of statistical

All 4 Chapters in Practical Work in Geography Part II