Data Processing - Class 12 Geography - Chapter 2 - Notes, NCERT Solutions & Extra Questions
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Extra Questions - Data Processing | Practical Work in Geography | Geography | Class 12
Consider the frequency distribution of the given numbers.
Value | 1 | 2 | 3 | 4 |
Frequency | 5 | 4 | 6 | f |
If the mean is known to be 3, then the value of $f$ is: A) 3 B) 7 C) 10 D) $\quad 14$
To determine the value of $ f $, we start by using the formula for the mean of a frequency distribution:
$$ \text{Mean} = \frac{\sum x_i f_i}{\sum f_i} $$
Given that the mean is 3, we set up the following equation based on the provided data:
$$ 3 = \frac{1 \times 5 + 2 \times 4 + 3 \times 6 + 4 \times f}{15 + f} $$
Calculate the numerator (sum of products of values and frequencies):
$$ 1 \times 5 + 2 \times 4 + 3 \times 6 + 4 \times f = 5 + 8 + 18 + 4f = 31 + 4f $$
So the equation becomes:
$$ 3 = \frac{31 + 4f}{15 + f} $$
To find $ f $, cross-multiply to clear the fraction:
$$ 3(15 + f) = 31 + 4f $$
Expand and simplify:
$$ 45 + 3f = 31 + 4f $$
Rearrange to isolate $ f $:
$$ 45 - 31 = 4f - 3f \ 14 = f $$
Thus, the value of $ f $ is 14.
Therefore, the correct option is D) 14.
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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
(i) (d) Median - The median, being a positional measure, does not get affected by extreme values since it simply represents the middle value in a data set.
(ii) (d) Mode - The mode is the value that appears most frequently in a data set and is always located at the hump (peak) of the distribution.
Define the mean. (30 words)
Mean is the arithmetic average of a data set, calculated by summing all values and dividing the total by the number of observations. It represents the central value of the data.
What are the advantages of using mode ? (30 words)
Mode is advantageous because it is easy to understand and calculate, represents the most frequent score in a dataset, and is especially useful for categorical or nominal data. It helps identify the most common category.
Explain relative positions of mean, median and mode in a normal distribution and skewed distribution with the help of diagrams. (125 words)
In a normal distribution, the mean, median, and mode are all located at the same point on the distribution curve, symbolizing perfect symmetry. This distribution is visually represented as a bell-shaped curve (Figure 2.3), where most data points cluster around the central peak, and the frequencies gradually decrease symmetrically as the values move away from the center.
In skewed distributions, the positions of mean, median, and mode differ:
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Positive Skew (Figure 2.4): The tail on the right side is longer. The mode occurs at the highest frequency, the median is positioned further right, and the mean is pulled to the rightmost by extreme values.
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Negative Skew (Figure 2.5): The tail on the left side is longer. The mode occurs at the peak frequency, median a bit to its left, and the mean is further left influenced by the low outliers.
These diagrams visually illustrate how the relative positions of these measures of central tendency shift depending on the distributionโs symmetry or asymmetry.
Comment on the applicability of mean, median and mode (hint: from their merits and demerits). (125 words)
Mean is suitable for quantitative data and provides a precise measurement central tendency, but it is sensitive to outliers. It is highly applicable in reliable datasets without extreme values.
Median is robust against outliers, making it more suitable for skewed distributions. It determines the midpoint of a dataset, thereby representing the dataโs center accurately when extremities distort the mean. Applicability is primarily in income or property valuations, where extremes can skew data.
Mode is applicable for qualitative and quantitative data, identifying the most frequent occurrence. This measure is simplest to understand, making it particularly useful in consumer behavior analysis for identifying common preferences or behaviors. However, its utility might be limited if no single value repeats or if data is uniformly distributed.
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Comprehensive Data Processing Class 12 Notes
Introduction to Data Processing
Data processing is a crucial component of the class 12 curriculum because it helps in organising and presenting data to make it comprehensible. This involves various statistical techniques that facilitate the analysis of data, such as Measures of Central Tendency, Measures of Dispersion, and Measures of Relationship.
Measures of Central Tendency
Explanation and Types
Measures of central tendency are statistical techniques used to summarise a set of data with a single value that represents the centre of its distribution. The three main types of central tendency measures are the mean, median, and mode.
Importance in Data Processing
These measures help in simplifying complex datasets by providing a clear indication of the 'central' value around which other data points cluster, making it easier to understand and interpret the data.
Mean - A Measure of Central Tendency
Computing Mean from Ungrouped Data
Direct Method
The mean is derived by summing all the data points and dividing by the number of observations. For ungrouped data, the direct method formula is:
[ \overline{\mathrm{X}} = \frac{\sum x}{N} ]
Example: Let's say you have rainfall data from seven districts.
- Sum all the rainfall data.
- Divide by the number of districts.
[ \overline{\mathrm{X}} = \frac{6484}{7} = 926.29 ]
Indirect Method
For large datasets, the indirect method is more efficient. It involves coding the data to reduce values by subtracting a constant known as the assumed mean.
[ \overline{\mathrm{X}} = A + \frac{\sum d}{N} ]
Where ( A ) is the subtracted constant (assumed mean), and ( \sum d ) is the sum of coded scores.
Example: Using the same rain data with an assumed mean of 800:
[ \overline{\mathrm{X}} = 800 + \frac{884}{7} = 926.29 ]
Computing Mean from Grouped Data
Grouped data means working with frequency distributions rather than individual data points.
Direct Method
For grouped data, the mean is calculated using midpoint values of class intervals:
[ \overline{\mathrm{X}} = \frac{\sum f x}{N} ]
Example:
Class intervals grouped data with 99 observations, as shown below:
[ \overline{\mathrm{X}} = \frac{10160}{99} = 102.6 ]
Indirect Method
This method simplifies computations by coding:
[ \overline{\mathrm{X}} = A \pm \frac{\sum f d}{N} ]
Where ( A ) is the midpoint of the assumed mean group.
Example:
Assumed mean ( A ) is 100 from the midpoint of group 90-110:
[ \overline{\mathrm{X}} = 100 + \frac{260}{99} = 102.6 ]
Median - A Positional Average
Computing Median for Ungrouped Data
When data is ungrouped, arrange it in ascending or descending order and locate the mid-point:
[ \text{Median} = \text{Value of } \left(\frac{N+1}{2}\right)^{\text{th}} \text{ item} ]
Example: For mountain peaks:
Given data - (8126, 8611, 7817, 8172, 8076, 8848, 8598)
Arranged data - (7817, 8076, 8126, 8172, 8598, 8611, 8848)
[ \text{Median} = 4^{\text{th }}\text{item} = 8172 ]
Computing Median for Grouped Data
Grouped data requires identifying the median class and using the formula:
[ M = l + \frac{i}{f}\left(\frac{N}{2} - c\right) ]
Where;
- ( M ) = Median
- ( l ) = Lower limit of the median class
- ( i ) = Class interval
- ( f ) = Frequency of median class
- ( c ) = Cumulative frequency of pre-median class
Example:
Class Distribution:
[ \text{Median class } = \frac{50}{2} = 25 \text{ (falls in group 80-90)} ]
[ M = 80 + \frac{10}{16} \times (25 - 21) = 82.5 ]
Mode - The Most Frequent Value
The mode is the value that appears most frequently in a dataset.
Computing Mode for Ungrouped Data
Arrange data to identify the most frequent value easily.
Example: Test scores:
Given data - (61, 10, 88, 37, 61, 72, 55, 61, 46, 22)
Arranged data - (10, 22, 37, 46, 55, 61, 61, 61, 72, 88)
Mode - (61 \text{ (appears 3 times)})
Computing Mode for Grouped Data
For grouped data, the modal class is identified and mode is computed accordingly.
Comparison of Mean, Median, and Mode
The three measures can be compared using a normal distribution curve.
Normal Distribution
In a normal distribution, the mean, median, and mode coincide.
Skewed Distributions
In skewed distributions, the mean, median, and mode do not coincide, affecting the data's central tendency.
graph LR
A[Normal Distribution]
B[Positively Skewed] --> C[Mean > Median > Mode]
D[Negatively Skewed] --> E[Mean < Median < Mode]
Understanding these differences helps in accurately interpreting data distributions.
Conclusion
Understanding measures of central tendency is vital for effective data processing in class 12. Mastering these concepts provides a solid foundation for further statistical analysis and data interpretation.
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