Q.Explain, with the help of a diagram, the properties of Lorenz curve. Or The number of members of six families are respectively: 10, 9, 10, 9, 9, 10. Find the standard deviation of the number of members of these families.

Standard deviation is defined as the positive square root of the average squared deviation from the mean, so by construction it can never be negative — it is always positive (or zero when there is no variability at all).

Standard deviation (SD) is computed as:

SD = square root of [ Σ(X − mean)² / N ]

Every individual deviation (X − mean) is squared before being averaged, which eliminates the sign (a negative deviation squared becomes positive, same as a positive deviation squared). The sum of these squared terms, divided by N, is therefore always zero or positive, and by mathematical convention we take only the positive (principal) square root as the standard deviation, never the negative root.

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'SD depends on all values' — True, since every observation enters the sum of squared deviations. 'Range stays the same if every value is multiplied by 10' — False, range is also multiplied by 10.

Main statement: Standard deviation is calculated as the square root of the average of the squared deviations of EVERY value from the mean: SD = √[Σ(X − mean)² / N]. Because each and every observation contributes a term to this sum (not just the extreme values), standard deviation genuinely reflects the spread of the complete data set. This statement is True.

Or alternative: Range = Maximum value − Minimum value. If every value in the data set is multiplied by a constant k (here, k = 10), the new maximum becomes 10 × (old maximum) and the new minimum becomes 10 × (old minimum). The new range is therefore:

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Correction: "relative" should be absolute measure of dispersion.

Absolute measures of dispersion are in the original units of the data; standard deviation $\sigma$ is one such absolute measure. The corresponding relative measure (unit-free) is the coefficient of …

Variance is a more refined (improved) measure of dispersion than the simple range, since it is based on every value's deviation from the mean rather than just the two extreme values; the Lorenz ratio's maximum value is 1, occurring under complete/perfect inequality.

Main blank -- 'Variance is the ___ of the range': The range only uses the highest and lowest values of a series and ignores every value in between, so it can be misleading when there are extreme outliers or when most of the data is bunched together. Variance improves on this by using the deviation of EVERY observation from the arithmetic mean (squared, then averaged), so it reflects the dispersion of the whole series, not just its two extremes. Variance is therefore considered a more scientific/improved measure of dispersion compared with the range.

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Standard deviation is a measure of dispersion.

Standard deviation is defined as the positive square root of the mean of the squared deviations of the values from their arithmetic mean. It quantifies how widely the observations are spread (scattered) about the mean — that is exactly what a measure of dispersion does. Measures of central tendency (mean …

Replace relative with absolute: standard deviation is an absolute measure of dispersion.

Measures of dispersion are of two kinds:

• Absolute measures — expressed in the same units as the data (range, quartile deviation, mean deviation, standard deviation).
• Relative measures — unit-free ratios/percentages used for comparison (coefficient of range, coefficient of variation, etc.). …

$\bar{x}=3$, $\sigma=\sqrt{\frac{(-1)^2+(1)^2}{2}}=1$.

Mean:

$\bar{x}=\frac{2+4}{2}=3.$

Deviations and their squares:

$2-3=-1\Rightarrow 1,\qquad 4-3=1\Rightarrow 1.$

Variance and standard deviation: …

The correct word is zero — equal values give zero dispersion.

Standard deviation measures how much the values scatter around their arithmetic mean:

$\sigma = \sqrt{\frac{\sum (x-\bar{x})^2}{N}}$

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A variable with no variation at all has standard deviation exactly zero, option (b).

Standard deviation measures the extent to which individual values of a variable are spread out around their mean. If every single observation in a dataset has the SAME value (here, all equal to 2), then the mean of the dataset is also exactly 2, and every individual observation's DEVIATION from the mean (xi - mean) is 0. Since standard deviation is the square root of the average of these squared deviations, and every squared deviation here is 0, the standard deviation itself must be exactly 0. Standard deviation, being a square root of a sum of squares (which can never be negative), can never itse …

A higher Gini coefficient means greater income inequality; standard deviation's minimum possible value is zero.

Main blank - Gini coefficient

The Gini coefficient is a widely-used summary measure of income (or wealth) inequality, derived from the Lorenz curve, and ranges in value from 0 to 1. A value of 0 represents PERFECT EQUALITY (everyone has an identical income), while a value of 1 represents MAXIMUM/complete inequality (one person/household holds the entire income of the economy). Therefore, the GREATER the value of the Gini coefficient, the GREATER is the degree of inequality in the distribution of income in that economy - this is exactly how the measure is constructed and interpreted.

Or - Minimum value of standard deviation …

Zero income inequality makes the Lorenz curve coincide with the diagonal; identical data values always give a standard deviation of zero.

Lorenz curve and zero inequality. The diagonal line in a Lorenz-curve diagram represents perfect equality — every X% of the population holding exactly X% of total income. The ACTUAL Lorenz curve coincides with this diagonal only in the special case where income inequality is ZERO, i.e., income is distributed perfectly equally among the population.

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