NCERT Solutions For Class 12 Geography Part C Chapter 2 Data Processing
Q. 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
- Ans. (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
- Ans. (b) Median and Mode
(iii) A scatter plot represents negative correlation if the plotted values run from:
- (a) Upper left to lower right
- (b) Lower left to upper right
- (c) Left to right
- (d) Upper right to lower left
- Ans. (a) Upper left to lower right
Q. Answer the following questions in about 30 words:
(i) Define the mean.
Ans. The mean is the result of adding all the values together and dividing by the number of observations.
$$\bar{\text{X}}=\frac{\text{Sum of observations}}{\text{No. of observations}}$$
(ii) What are the advantages of using mode ?
Ans. The highest occurrence or frequency at a certain location or value is known as the mode. The most significant benefit of the mode is that it is unaffected by extreme values. Even open-ended series can be defined.
(iii) What is dispersion ?
Ans. The scattering of scores around the measure of central tendency is referred to as ‘dispersion.’ It’s a metric for determining how much individual items or numerical data fluctuate or spread around an average value. As a result, we must employ a measure of central tendency and dispersion or variability to better understand a distribution.
(iv) Define correlation.
Ans. Correlation is a measure of how closely two or more sets of data are related. It has a very useful function.
(v) What is perfect correlation ?
Ans. Ans. A perfect correlation indicates that two variables have a proportionate connection. It’s a perfect positive connection if doubling x also doubles the value of y. On the other hand, perfect negative correlation occurs when the value of y is half when the variable x is doubled.
(vi) What is the maximum extent of correlation ?
Ans. The maximum degree of correspondence or relation goes upto 1 (one) in mathematical terms.
Q. 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.
Ans. (a) Normal Curve: The maximum frequency occurs in the centre of this curve, and both the left and right tails are identical. It’s a unimodal curve with the same mean, median, and mode. A bell shaped or symmetrical curve is another name for it. It is as follows:
(b) Positively Skewed Curve: This is a symmetrical curve with a tail on the right hand side of the graph and higher frequencies for lower data values. The curve is on the left side of the distribution in these histograms. The majority of the distribution is concentrated on the left if the right tail is longer. It has a small number of low values. It is as follows:
(c) Negatively Skewed Curve: This is a symmetrical curve with a tail on the left side of the graph and greater frequencies for higher data values. The left tail is longer, and the distribution’s bulk is concentrated on the figure’s right side. It has a small number of low values. The distribution is biassed to the left. It is as follows:
(ii) Comment on the applicability of mean, median and mode (hint: from their merits and demerits).
Ans. Mean:
- 1. It is the most basic of all the central tendency measurements.
- 2. It is based on the entire sequence of things. As a result, it represents the value of various items.
- 3. It is a value. There is no provision for estimated values.
- 4. It is a type of core tendency that is stable.
- 5. It can be used as a comparison.
Median:
- 1. The extreme values in the series have little effect on the median.
- 2. Only the middle values and their units are needed to get the median.
- 3. Data can also be represented graphically to find the median.
- 4. In a series, the median value is always known.
Mode:
- 1. The mode is a simple measure of central tendency.
- 2. Extreme and marginal values have less of an impact.
- 3. The finest portrayal of the series is mode.
- 4. It may be calculated graphically as well.
(iii) Explain the process of computing Standard Deviation with the help of an imaginary example.
Ans. Ans. The most often used metric of dispersion is the standard deviation (SD). It is defined as the square root of the average of squares of deviations. It’s always measured as a percentage of the mean. The standard deviation is the most consistent measure of variability and is utilised in a wide range of statistical procedures.
Steps:
- 1. To calculate SD, first square the departure of each score from the mean (x) that is (x^{2}).
- 2. Above step turns all negative deviation indications into positive ones. It protects SD from the primary criticism levelled at mean deviation when modulus x is used. The squared variances are then added together x^{2}.
- 3. The square root is obtained by dividing the total of the squared deviations (x<sup>2</sup>) by the number of instances. As a result, the root means square deviation is used to determine Standard Deviation.
- Lets take an example for an ungrouped class: Calculate the standard deviation for the following scores:
Steps to be followed:
X | $$x(\text{X}-\bar{\text{X}})$$ | x^{2} |
1 | -4 | 16 |
3 | -2 | 4 |
5 | 0 | 0 |
7 | 2 | 4 |
9 | 4 | 16 |
$$\Sigma X = 25\\\text{}\\ \text{N}=5\\\text{}\\ \therefore \space\space = 5\\ \sigma=\frac{\sqrt{\Sigma x^{2}}}{\text{N}}\\=\frac{\sqrt{40}}{5}=\sqrt{8}=2.828\\=2.83\space\text{(round of F).}$$
(iv) Which measure of dispersion is the most unstable statistic and why?
- Ans. The most unstable statistic is a range because:
- 1. The range does not include all words. Only the most extreme products accurately portray its magnitude. As a result, the range cannot fully represent the data since all other middle values are ignored.
- 2. The range is not a good measure of dispersion for the reasons stated above.
- 3. Even if all other phrases and variables in between are modified, the range remains unchanged.
- 4. Variation in a sample has a significant impact on range. The range varies from one sample to the next. As the sample size grows, so does the range, and vice versa.
- 5. It provides little insight into the variability of other data.
- 6. The range of open-end intervals is indetermined because the lower and appearing bounds of the first and final intervals are not stated.
(v) Write a detailed note on the degree of correlation.
Ans. We can assess the degree or extent of the connection between the two variables using the coefficient of correlation. We can also establish if the connection is positive or negative, as well as its degree or extent, using the coefficient of correlation. A complete positive correlation exists
when two variables change in the same direction and in the same proportion. The coefficient of correlation in this example, according to Karl Pearson, is +1. If the variables vary in the same proportion and in the opposite direction, the correlation is perfect negative. It has a correlation coefficient of –1.
- 1. Absence of correlation: If two series of two variables show no relationship between them, or if a change in one variable does not result in a difference in the other, we can confidently conclude that there is no connection or ludicrous correlation between the two variables. In this example, the correlation coefficient is 0.
- 2. Limited degrees of correlation: We call a correlation Limited if two variables are not fully connected or if there is a perfect lack of association. It might be positive, negative, or zero, but it must be within the range of 1.
- This type of association is divided into three categories: high, moderate, and low.
(vi) What are various steps for the calculation of rank order correlation?
- Ans. The steps to calculate the rank order correlation are as follows:
- Step 1: Sort the data into two groups. Give the highest value number one, the second highest number two, and so on.
- Step 2: Determine the ranks’ differences, d.
- Step 3: Calculate the difference squares (d^{2}).
- Step 4: Add these squared differences together to get Σd^{2}.
- Step 5: Spearman’s Rank Correlation Substituting this sum into the following formula yields the coefficient.
$$1-\frac{6×Σd^{2}}{n(n^{2}-1)}$$
where n is how many pairs of data you have.