CBSE Class 11 Maths Chapter Notes: Statistics

Key Statistical Terms

Measures of Dispersion

Quantities that describe how spread out or scattered a set of data is from its central value (like mean or median).

Range

The difference between the maximum and minimum values in a data set. Simplest measure of dispersion.

Mean Deviation

The average of the absolute differences between each data point and the mean or median of the data set.

Variance

The average of the squared differences from the mean. It measures how far each number in the set is from the mean and therefore from every other number in the set.

Standard Deviation (SD)

The square root of the variance. It indicates the typical distance between a data point and the mean of the data set. Most widely used measure of dispersion.

Coefficient of Variation (CV)

A relative measure of dispersion, expressed as a percentage. It is the ratio of the standard deviation to the mean, multiplied by 100. Used to compare variability between different data sets.

Frequency Distribution

A table that shows how often each value (or range of values) appears in a data set. Can be discrete or continuous.

Understanding Measures of Dispersion

Measures of dispersion (or variability) tell us how spread out the data points are. While measures of central tendency (mean, median, mode) describe the center of the data, they don't tell us about its distribution. For example, two different datasets can have the same mean but vastly different spreads.

• Range is the simplest measure, calculated as the difference between the highest and lowest observations. It's easy to compute but highly affected by extreme values and only considers two data points, making it a crude measure.

• Quartile Deviation (not extensively covered in CBSE Class 11, but good to know for context) is based on quartiles and measures the spread of the middle 50% of the data, making it less affected by extreme values than the range.

• Mean Deviation (MD) measures the average of the absolute deviations from a central value (mean or median). Its advantage is that it considers all observations. However, using absolute values makes it mathematically less tractable for further analysis compared to variance or standard deviation. MD about the median is generally less than or equal to MD about the mean.

• Variance is the average of the squared deviations from the mean. Squaring the deviations eliminates the problem of positive and negative differences cancelling out. It gives more weight to larger deviations. Variance is extensively used in inferential statistics.

• Standard Deviation (SD) is the square root of the variance. It's the most widely used measure because it brings the unit of dispersion back to the original unit of the data, making it more interpretable than variance. A low standard deviation indicates that data points are generally close to the mean, while a high standard deviation indicates that data points are spread out over a wider range. Effect of Change of Origin and Scale: SD and Variance are independent of the change of origin but are affected by the change of scale. If each observation is multiplied by a constant 'c', the new standard deviation becomes |c| times the original standard deviation, and the new variance becomes c² times the original variance.

Essential Formulas for Measures of Dispersion

Steps to Calculate Standard Deviation for Grouped Data (Short-cut Method)

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Worked Examples

• Example 1: Mean Deviation about Mean (Ungrouped Data)

  Q: Find the Mean Deviation about the mean for the data: 6, 7, 10, 12, 13, 4, 8, 12.

  A:

  1. Calculate Mean ($\bar{x}$): $\bar{x} = \frac{6+7+10+12+13+4+8+12}{8} = \frac{72}{8} = 9$
  2. Calculate absolute deviations $|x_i - \bar{x}|$:

  $|6-9|=3, |7-9|=2, |10-9|=1, |12-9|=3, |13-9|=4, |4-9|=5, |8-9|=1, |12-9|=3$

  1. Sum of absolute deviations: $\sum |x_i - \bar{x}| = 3+2+1+3+4+5+1+3 = 22$
  2. Mean Deviation: $MD(\bar{x}) = \frac{22}{8} = 2.75$

• Example 2: Variance and Standard Deviation (Ungrouped Data)

  Q: Find the Variance and Standard Deviation for the data: 3, 5, 7, 9, 11.

  A:

  1. Calculate Mean ($\bar{x}$): $\bar{x} = \frac{3+5+7+9+11}{5} = \frac{35}{5} = 7$
  2. Calculate $(x_i - \bar{x})^2$:

  $(3-7)^2 = 16$
  $(5-7)^2 = 4$
  $(7-7)^2 = 0$
  $(9-7)^2 = 4$
  $(11-7)^2 = 16$

  1. Sum of squared deviations: $\sum (x_i - \bar{x})^2 = 16+4+0+4+16 = 40$
  2. Variance ($\sigma^2$): $\sigma^2 = \frac{40}{5} = 8$
  3. Standard Deviation ($\sigma$): $\sigma = \sqrt{8} \approx 2.83$

Key Points to Remember

• Measures of dispersion quantify the spread or variability of data.
• Range is simplest but most affected by extreme values.
• Mean deviation considers all observations but uses absolute values, limiting its mathematical utility.
• Variance and Standard Deviation are the most robust and widely used measures of dispersion.
• Standard Deviation (SD) has the same unit as the original data, making it easy to interpret.
• SD is independent of change of origin (adding/subtracting a constant) but depends on change of scale (multiplying/dividing by a constant). If each observation is multiplied by 'c', SD is multiplied by |c|.
• Coefficient of Variation (CV) is a unitless relative measure used for comparing variability of two different datasets.
• Always remember to take the positive square root for Standard Deviation.
• For grouped data, $x_i$ represents the mid-point of the class interval.

Exam Tip: Avoiding Common Traps

When calculating Mean Deviation, remember to take the absolute value of the differences $|x_i - \bar{x}|$ or $|x_i - M|$. A common mistake is to forget the absolute value, which will always result in a sum of zero for deviations from the mean.

For Variance and Standard Deviation, ensure you correctly calculate the mean ($\bar{x}$) first, as all subsequent calculations depend on it. Double-check your squares and sums. When dealing with grouped data, ensure you use the mid-points ($x_i$) of the class intervals correctly and multiply by the respective frequencies ($f_i$) where required. Pay close attention to whether the question asks for variance, standard deviation, or coefficient of variation. Each has a distinct formula and interpretation.

Practice Questions with Solutions

• Q: What is the primary difference between variance and standard deviation?
  A: Variance is the average of the squared deviations from the mean, while standard deviation is the square root of the variance. SD is in the same unit as the data, making it more interpretable.

• Q: If each observation in a dataset is increased by 5, how does the standard deviation change?
  A: The standard deviation remains unchanged. It is independent of the change of origin (addition or subtraction of a constant).

• Q: When is Coefficient of Variation (CV) most useful?
  A: CV is most useful when comparing the variability of two different datasets, especially if they have different units or widely different means. It provides a relative measure of dispersion.

• Q: What is the formula for mean deviation about the median for grouped data?
  A: The formula is $MD(M) = \frac{\sum f_i |x_i - M|}{\sum f_i}$, where M is the median, $x_i$ are mid-points, and $f_i$ are frequencies.

Frequently Asked Questions