Taking the root to get the spread back into the units of the data

What to assume they know

• Squaring removes the sign and punishes the far-out values — the variance, and why the plain total of squares would not do
• That the square root of a non-negative number has a non-negative value by convention
• That squaring and taking a square root preserve order among non-negative numbers
• Units carrying through an arithmetic operation
• The step-deviation route to a mean, from The same procedure once the data arrive already grouped

What they should be able to do

• State why the variance cannot be compared with the mean or plotted on the observations' own axis
• Define the standard deviation as the non-negative square root of the variance, and say why the non-negative root is chosen
• Explain why taking the root cannot change which of two data sets is judged more dispersed
• Compute the variance of an ungrouped data set with the mean found by the step-deviation method
• Compute the corresponding standard deviation and round it as the chapter does
• Compare the standard deviation, the mean deviation and the range of the same data set and say what each is telling you
• Identify a printed slip in the chapter's own working table and say what the column should have read

Where it usually goes wrong

• "Variance and standard deviation are two different measures." They are the same measure in two unit systems, and either determines the other. Choose by what you need to do with it: report the standard deviation, compute with the variance.
• "The square root can be negative." By convention the root written here is the non-negative one, because a standard deviation is a distance. The chapter says so in the sentence that introduces it.
• "Taking the root might reverse which data set is more spread." It cannot. The square root is increasing on the non-negative numbers.
• "The step-deviation columns can also be used for the squares." They cannot. The step-deviations in Table 13.7 are measured from 14 and shrunk by 2; the squared column is measured from the true mean, 15, at full scale. What happens to a variance under shifting and scaling is the subject of §13.5.4, not of a shortcut taken here.
• "5.74 is exact." It is the square root of 33 rounded to two places. Keep the surd until the last line.
• "A standard deviation of 5.74 means most observations are within 5.74 of the mean." Nothing in this chapter says that. It is a typical squared-weighted distance, not a guarantee about how many observations fall inside it. For this data four of the ten lie further out than 5.74.
• "Both mean deviation and standard deviation should give the same number." For the same data they generally differ, and the standard deviation is the larger unless every distance from the mean is identical.

Questions to check understanding

• Compute the variance and the standard deviation of a short ungrouped list
• Given a variance, write down the standard deviation, and the reverse
• State the units of the variance and of the standard deviation for data measured in a named unit
• Use an assumed mean and a common factor to find the mean, then complete the variance from the true deviations
• Explain why the standard deviation of a data set is never less than its mean deviation about the mean
• Report a square root to two decimal places

Examples worth working on the board

Values marked verified are worked out here on data printed in this chapter.

• The units argument (§13.5.1, p. 274). The chapter's reason is short: a variance is built from squares, so it does not share units with the observations or with their mean. Make it concrete by carrying an actual unit through — the batting records of §13.1 are in runs, so their variances are in runs-squared, and 1300.6 runs-squared cannot be marked on the same axis as a mean of 53 runs. The square root of 1300.6 is about 36.1, and 36.1 runs can be.
• The definition (§13.5.1, p. 274). Standard deviation is the non-negative square root of the variance, written with a plain sigma. The chapter labels this displayed line (1) and refers back to that number later. Two things worth saying aloud: the root is taken to be non-negative so that the measure is a distance, and the variance is never negative, so the root always exists.
• Order is preserved. For non-negative numbers, if one is larger than another then so is its square root. So any ranking of data sets by variance is the same ranking by standard deviation. This is why the two are not competing measures — they are one measure in two unit systems. The chapter does not make this point; it is one line and it prevents a great deal of confusion.
• Example 8 (§13.5.1, pp. 274–275, Table 13.7). Data: 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 — ten observations, evenly spaced two apart. The chapter works the mean by the step-deviation method with assumed mean 14 and common factor 2. Verified: the step-deviations are −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, totalling 5; the mean is therefore 14 + (5 ÷ 10) × 2, which is 15.
• The deviation column. Verified: the deviations from the true mean 15 are −9, −7, −5, −3, −1, 1, 3, 5, 7, 9 — the odd numbers, paired either side. Their squares are 81, 49, 25, 9, 1, 1, 9, 25, 49, 81, totalling 330, which is twice 1 + 9 + 25 + 49 + 81. Verified: the variance is 330 ÷ 10 = 33, and the standard deviation is the square root of 33, which the chapter reports as 5.74. Note: the mean, 15, is not one of the ten observations — they are all even — so no deviation is zero.
• A printed slip in Table 13.7 (p. 274). The fourth column is headed with a deviation from the mean and no exponent, although its entries are the squares — 81 against a deviation of −9, and so on down. The third column, headed the same way, carries the deviations themselves. Show the column correctly headed and say the printed heading is short of a square; a student following the book will otherwise think the two columns should agree.
• Three measures of the same ten numbers (not in the book). Verified: the range is 24 − 6 = 18; the mean deviation about the mean is 50 ÷ 10 = 5; the standard deviation is about 5.74. All three are in the units of the data, and they can be marked on one axis beside the mean of 15. The range reports the whole span, the mean deviation the typical distance, the standard deviation the same typical distance with the outer observations weighted up.
• Why the standard deviation here exceeds the mean deviation. Verified: 5.74 against 5 for the same ten numbers. This is not an accident of this data: for any list of distances, the root of the mean of the squares is at least the mean itself, with the two equal only when all the distances are identical. State the condition — for this data the distances are 9, 7, 5, 3, 1 twice over and are not all equal, so the inequality is strict.

Figures to have open

• A number line 0 to 30 carrying the ten observations of Example 8, the mean at 15, and the three measures drawn as spans from the mean. An added figure; §13.5.1 prints no diagram.
• Table 13.7 redrawn with four columns, the fourth correctly headed as a squared deviation. The chapter's own table (p. 274) with the heading corrected.
• A units strip for section 2: observations in runs, deviations in runs, squares in runs-squared, mean of squares in runs-squared, root in runs. Standard schematic and the clearest single picture of the argument.
• No figure is printed between the definition on p. 274 and the end of Example 8 on p. 275; both pages carry text, one table and displayed formulas.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 13 "Statistics", §13.5.1 "Standard Deviation", printed pp. 274–275, including the displayed definition labelled (1) on p. 274 and Example 8 with Table 13.7.
• The variance this section takes the root of is defined at the top of p. 274, at the end of §13.5; that belongs to Squaring removes the sign and punishes the far-out values.
• The step-deviation method used in the table is set out in §13.4.2, pp. 266–268.