Mathematics · Class 11 Science
Ch 13Statistics — Class 11 Mathematics, concept-first.
Dispersion (also called variability or spread) measures how scattered or spread out the observations of a data set are around a central value such as the mean or median.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Variance and Standard Deviation of Data
For a data set these measure spread about the mean. For ungrouped data with mean , and the standard deviation is . For a frequency distribution with values (class midpoints for grouped data) and frequencies , ,
Most relevant Q&A
- Two batsmen A and B, in their last 5 innings, scored the following runs: | Batsman A | 40 | 50 | 60 | 45 | 55 | |---|---|---|---|---|---| |…Free
- Find the variance and standard deviation of the data: $6, 8, 10, 12, 14$.Preview
- Find the variance and standard deviation of the data: $4, 8, 6, 5, 9, 10, 3, 7$.Free
- Find the variance and standard deviation of the first 10 natural numbers: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.Free
- Find the variance and standard deviation of the data: $-4, -1, 0, 2, 5, 6, -2, 2$.Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction to Dispersion
Dispersion (also called variability or spread) measures how scattered or spread out the observations of a data set are around a central value such as the mean or median.
Range
Range is the simplest measure of dispersion: the difference between the largest (maximum) and the smallest (minimum) observation in a data set.
Mean Deviation about the Mean and Median (Ungrouped Data)
Mean deviation (also called mean absolute deviation) measures the average distance of the observations from a fixed central value — usually the mean or the median — treating every distance as a positi…
Mean Deviation for Grouped Data
For grouped data — where each distinct value (or each class interval, represented by its midpoint) carries an associated frequency — the mean deviation formulas of the previous section are adapted by…
Variance and Standard Deviation (Ungrouped Data)
Variance is another way of averaging the deviations of the observations from the mean while avoiding the identical-cancellation problem noted in the previous section — but instead of taking the absolu…
Variance and Standard Deviation for Grouped Data
The variance and standard deviation formulas extend to grouped data exactly as the mean deviation formulas did in the previous section — every squared deviation is weighted by the frequency with which…
Shortcut (Step-Deviation) Method for Grouped Data
When a continuous frequency distribution has midpoints that are inconveniently large or do not share a common factor, computing and directly by the methods of the previous two sections can involve unw…
Summary
This chapter developed four measures of how spread out a data set is, in increasing order of sophistication: - Range (ungrouped/discrete); for continuous data, the upper limit of the last class minus…
More questions
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- Example 1Find the range of the following data giving the daily maximum temperatures (in $^\circ$C) recorded in Kolkata over 10 days: $32, 35, 30, 33,…Free
- Example 2The marks obtained by 8 students in a class test are: $6, 7, 10, 12, 13, 4, 8, 12$. Find the mean deviation about the mean.Free
- Example 3Find the mean deviation about the median for the data: $13, 17, 16, 14, 11, 13, 10, 9, 15$.Preview
- Example 4Find the variance and standard deviation of the data: $6, 8, 10, 12, 14$.Preview
- Example 5Find the variance and standard deviation of the following discrete frequency distribution: | $x_i$ | 2 | 4 | 6 | 8 | 10 | |---|---|---|---|-…Preview
- Example 6Find the mean deviation about the mean for the following frequency distribution of the marks obtained by 50 students in a class test: | Clas…Preview
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- Q7Find the range of the marks obtained by 12 students in a unit test: $45, 52, 38, 61, 47, 55, 42, 58, 33, 49, 63, 40$.Free
- Q8The minimum temperatures (in $^\circ$C) recorded at a hill station over 7 days were: $2, -1, 4, 0, -3, 5, 1$. Find the range.Free
- Q9Find the mean deviation about the mean for the data: $10, 11, 7, 13, 12, 5, 9, 8, 14, 11$.Preview
- Q10Find the mean deviation about the median for the data: $6, 9, 3, 15, 12, 8, 10, 5$.Preview
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- Q11Find the mean deviation about the mean for the discrete frequency distribution: | $x_i$ | 5 | 10 | 15 | 20 | 25 | |---|---|---|---|---|---|…Free
- Q12Find the mean deviation about the mean for the following distribution of the ages (in years) of 50 members of a club: | Class interval | 10-…Preview
- Q13Find the mean deviation about the mean for the following distribution of marks obtained by 40 students in an examination: | Class interval |…Preview
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- Q14Find the variance and standard deviation of the data: $4, 8, 6, 5, 9, 10, 3, 7$.Free
- Q15Find the variance and standard deviation of the first 10 natural numbers: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.Free
- Q16Find the variance and standard deviation of the data: $-4, -1, 0, 2, 5, 6, -2, 2$.Free
- Q17The mean of 5 observations is $8$. If three of the observations are $5, 9, 10$ and the remaining two observations are equal, find the two eq…Preview
- Q18Find the variance and standard deviation of the discrete frequency distribution: | $x_i$ | 4 | 8 | 12 | 16 | 20 | |---|---|---|---|---|---|…Preview
- Q19Find the variance and standard deviation of the following distribution of the weights (in kg) of 30 students: | Class interval | 10-20 | 20-…Preview
- Q20Using the step-deviation method, find the variance and standard deviation of the following distribution of the marks obtained by 50 students…Preview