What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

Name the measures of dispersion studied.

Range, quartile deviation, mean deviation, variance and standard deviation (this chapter treats all except quartile deviation).

How is the range of a data set found?

Range = Maximum value − Minimum value.

Define mean deviation about the mean for ungrouped data.

\(\text{M.D.}(\bar x)=\dfrac{1}{n}\sum_{i=1}^{n}|x_i-\bar x|\), the mean of the absolute deviations from the mean.

Why are deviations squared in computing variance?

Absolute values are hard to handle algebraically; squaring makes all deviations non-negative and allows further algebraic treatment, giving the variance.

Define variance and standard deviation.

Variance \(\sigma^2=\dfrac{1}{n}\sum(x_i-\bar x)^2\); standard deviation \(\sigma=\sqrt{\sigma^2}\), expressed in the original units.

Write the frequency-distribution formula for standard deviation.

\(\sigma=\dfrac{1}{N}\sqrt{N\sum f_i x_i^2-\left(\sum f_i x_i\right)^2}\), where \(N=\sum f_i\).

What happens to the variance if each observation is increased by a constant?

It remains unchanged, since adding a constant shifts both the data and the mean equally.

If each observation is multiplied by \(k\), how does variance change?

The new variance is \(k^2\) times the original variance.

Find the mean deviation about the mean of \(6,7,10,12,13,4,8,12\).

Mean \(=9\); absolute deviations sum to \(22\), so \(\text{M.D.}=\dfrac{22}{8}=2.75\).

Why is standard deviation preferred over mean deviation?

Standard deviation uses squared deviations (avoiding absolute values), is amenable to further algebra and is the standard scientific measure of spread.

Name the three measures of central tendency.

The mean, the median and the mode.

Name the methods of finding the mean of grouped data.

The direct method, the assumed-mean (deviation) method and the step-deviation method.

Write the formula for the median of grouped data.

\(\text{Median} = l + \dfrac{\frac{n}{2} - cf}{f} \times h\), where \(l\) is the lower limit of the median class, \(cf\) the cumulative frequency before it, \(f\) its frequency and \(h\) the class size.

Write the formula for the mode of grouped data.

\(\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\), using the modal class and the frequencies around it.

State the empirical relationship between mean, median and mode.

\(\text{Mode} = 3 \times \text{Median} - 2 \times \text{Mean}\).

The mean of 5 numbers is 18. If one number is removed, the mean becomes 16. Find the removed number.

Sum of 5 numbers \(= 90\); sum of 4 numbers \(= 64\); so the removed number \(= 90 - 64 = 26\).

What is a cumulative frequency?

The running total of the frequencies up to a given class, used to draw an ogive and to find the median.

What does an ogive help to find?

The median of grouped data graphically, as the x-coordinate of the point where the two cumulative frequency curves meet (or where \(cf = n/2\)).

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Chapter 13: Statistics — Class 11 Mathematics

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The range of a data set is:

Summary

Measures of central tendency (mean, median, mode) tell where data are centred but not how spread out they are. This chapter studies measures of dispersion: range, mean deviation, variance and standard deviation. The range is simply maximum minus minimum. The mean deviation about a central value \(a\) (mean or median) is the average of the absolute deviations: for ungrouped data \(\text{M.D.}(\bar x)=\dfrac{1}{n}\sum|x_i-\bar x|\), with frequency-weighted versions for grouped data. Because absolute values are awkward to manipulate, variance squares the deviations: \(\sigma^2=\dfrac{1}{n}\sum (x_i-\bar x)^2\), and the standard deviation \(\sigma=\sqrt{\sigma^2}\) restores the original units. For a frequency distribution, \(\sigma=\dfrac{1}{N}\sqrt{N\sum f_i x_i^2-\left(\sum f_i x_i\right)^2}\), and a step-deviation (shortcut) method simplifies the arithmetic. Adding a constant to every observation leaves the variance unchanged, while multiplying by \(k\) multiplies the variance by \(k^2\). These measures let us compare the consistency of two data sets with the same mean.

Measures of dispersionRangeMean deviation for ungrouped and grouped dataVariance and standard deviationShortcut (step-deviation) methodEffect of changing observations on variance

Key terms

Dispersion

The scatter or spread of data about a central value.

Range

Maximum value minus minimum value of the data.

Mean deviation

Average of the absolute deviations from a central value \(a\): \(\dfrac{1}{n}\sum|x_i-a|\).

Variance

Mean of the squared deviations from the mean: \(\sigma^2=\dfrac{1}{n}\sum(x_i-\bar x)^2\).

Standard deviation

The positive square root of the variance, \(\sigma=\sqrt{\sigma^2}\).

Effect of change

Adding a constant leaves variance unchanged; multiplying by \(k\) multiplies variance by \(k^2\).

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Dispersion

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The scatter or spread of data about a central value.

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