Math on Info: A James B. Bleeker Website: Statistics 101: Norms, Spreads, Normality, and Chebyshev's Rule

13. An ordered set is denoted by parentheses (). For example, (5.4, 3.1, 5.4, 2.2). In an ordered set, duplicates may appear; here, we see 5.4 twice. Ordered sets are often used to list observed data.

14. Parentheses () are also used to denote open intervals of real numbers, i.e., intervals of real numbers that do not include the endpoints. Examples are:

Open intervals may also be denoted using the notation a < x < b or x > a. The above examples in this notation are:

Note that (1,) and 1 < x <

and 1 < x and x > 1 all denote the same interval (all real numbers greater than 1). All four notations are used frequently.

We begin with a quantitative data set (x₁, x₂, x₃, x₄, ..., x_n). Then we have:

19. a.
b.
c.
d.	...

This notational definition for summation generalizes rather immediately. Occasionally, the index values are dropped from the summation notation, for example,

in lieu of that found to the left of the equals sign in 15.a. above.

Note that the population standard deviation, variance, and mean employ Greek letters; the sample standard deviation and variance employ their Latin counterparts and the sample mean is designated by

The mean and median are values of data set centrality or middle-ness. When the chart is symmetrical with respect to a vertical line passing through the mean, the median equals the mean. As symmetry declines and the chart becomes more skewed to the right or left, the median diverges from the mean, as may be seen in the following histogram.

The variance and the standard deviation are measures of how spread out the data are. The standard deviation is a very commonly used measure of data spread.

Note that in the computation of the sample variance (and consequently the sample standard deviation), m-1 is used in the denominator in lieu of m, as it has been found to provide the better guide as to the population's actual standard deviation.

There is a more simple computation of the sample variance, and consequently the sample standard deviation, as the following computations show.

The z-score z_i, corresponding to a data set item x_i, measures in standard deviations the distance that x_i falls above the mean (positive) or below the mean (negative) of the population or sample. More generally, for any given value z, the z-score is the distance in standard deviations above the mean (positive) or below the mean (negative), and consequently the z-score of the mean is always 0. The z-score is an oft used measurement in statistics, as (1) it provides a uniform measure as to where a data value rests relative to the mean and (2) to it a specific percentage of the population or sample may often be attached (more on this follows immediately below).

In nature, often measurements of various traits of a population (say, wing width of a bird species) are distributed such that the resulting histogram for the population, or larger sample, closely approximates the common bell curve (normal curve), a curve (depicted below) that has perfect symmetry about a central axis and other well-defined mathematical properties. When the histogram of a population trait closely approximates the normal curve, the mean and median are coincident (or nearly so) and between the mean and any given z-score z there falls a percentage of the population that closely approximates the percentage of the total area under the normal curve that falls above the interval (0,z). (This percentage of the population is also approximated by the percentage of the total area under the frequency or relative frequency histogram that falls above the interval (0,z).)

In the two charts that follow, the percentages of the total area under the normal curve that fall above the intervals (0,1) and (0,2) are provided. When the histogram of a population's trait is bell-curve shaped, these percentages approximate the percentages of the population whose trait measurements fall between the mean and one standard deviation (z=1) and the mean and two standard deviations (z=2), respectively.

By symmetry, the percentage of the area under the normal curve and between the mean and z=-1 is 34.1% and that under the normal curve and between the mean and z=-2 is 47.7%. And consequently, the percentage of the area under the normal curve and between z=-1 and z=1 is 68.2% and the percentage of the area under the normal curve and between z=-2 and z=2 is 95.4%.

As the histograms of many population traits are bell-shaped and approximate the normal curve, tables giving the percentages of the total area under the normal curve that fall above specified z-score intervals have been assembled. One such a table appears below.

Table I: Table Giving the Percentages of the Total Area under the Normal Curve That Fall above the z-score Intervals (0,z)

z to one digit to the right of the decimal point

z's second digit to the right of the decimal point

0.0

0.0%

0.4%

0.8%

1.2%

1.6%

2.0%

2.4%

2.8%

3.2%

3.6%

0.1

4.0%

4.4%

4.8%

5.2%

5.6%

6.0%

6.4%

6.7%

7.1%

7.5%

0.2

7.9%

8.3%

8.7%

9.1%

9.5%

9.9%

10.3%

10.6%

11.0%

11.4%

0.3

11.8%

12.2%

12.6%

12.9%

13.3%

13.7%

14.1%

14.4%

14.8%

15.2%

0.4

15.5%

15.9%

16.3%

16.6%

17.0%

17.4%

17.7%

18.1%

18.4%

18.8%

0.5

19.1%

19.5%

19.8%

20.2%

20.5%

20.9%

21.2%

21.6%

21.9%

22.2%

0.6

22.6%

22.9%

23.2%

23.6%

23.9%

24.2%

24.5%

24.9%

25.2%

25.5%

0.7

25.8%

26.1%

26.4%

26.7%

27.0%

27.3%

27.6%

27.9%

28.2%

28.5%

0.8

28.8%

29.1%

29.4%

29.7%

30.0%

30.2%

30.5%

30.8%

31.1%

31.3%

0.9

31.6%

31.9%

32.1%

32.4%

32.6%

32.9%

33.1%

33.4%

33.6%

33.9%

1.0

34.1%

34.4%

34.6%

34.8%

35.1%

35.3%

35.5%

35.8%

36.0%

36.2%

1.1

36.4%

36.6%

36.9%

37.1%

37.3%

37.5%

37.7%

37.9%

38.1%

38.3%

1.2

38.5%

38.7%

38.9%

39.1%

39.3%

39.4%

39.6%

39.8%

40.0%

40.1%

1.3

40.3%

40.5%

40.7%

40.8%

41.0%

41.2%

41.3%

41.5%

41.6%

41.8%

1.4

41.9%

42.1%

42.2%

42.4%

42.5%

42.6%

42.8%

42.9%

43.1%

43.2%

1.5

43.3%

43.4%

43.6%

43.7%

43.8%

43.9%

44.1%

44.2%

44.3%

44.4%

1.6

44.5%

44.6%

44.7%

44.8%

44.9%

45.1%

45.2%

45.3%

45.4%

1.7

45.5%

45.6%

45.7%

45.8%

45.9%

46.0%

46.1%

46.2%

46.3%

1.8

46.4%

46.5%

46.6%

46.7%

46.8%

46.9%

47.0%

47.1%

1.9

47.1%

47.2%

47.3%

47.4%

47.5%

47.6%

47.7%

2.0

47.7%

47.8%

47.9%

48.0%

48.1%

48.2%

2.1

48.2%

48.3%

48.4%

48.5%

48.6%

2.2

48.6%

48.7%

48.8%

48.9%

2.3

48.9%

49.0%

49.1%

49.2%

2.4

49.2%

49.3%

49.4%

2.5

49.4%

49.5%

2.6

49.5%

49.6%

2.7

49.7%

2.8

49.7%

49.8%

2.9

49.8%

49.9%

3.0

49.9%

Note: The unit of measure of the z-score is standard deviation.

Sources: Statistics for Business and Economics: Eighth Edition, p. 987, and Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, pp. 966-972.

32. A population trait that has a histogram that is bell-shaped is often referred to as a bell-shaped distributed trait.

Note that by these definitions and the symmetry of the normal curve, we have, for a population with a bell-shaped distributed trait, %_p(z₁,z₂)

%_a(z₁,z₂), %_p(0,)

%_a(0,) = 50%, and %_p(-,0)

%_a(-,0) = 50%.

Irrespective of whether the histogram of a data set resembles a normal curve or resembles it not in the least, we still have Chebyshev's Rule: For any shaped histogram and for any z-score z > 1, the percentage of the population or sample with z-scores within the open interval (-z,+z) is greater than or equal to (1 - 1/z²)

(100%).