NDA · Chapter

Statistics

The concepts NDA needs, one connected idea at a time.

0 / 20

concepts mastered

Data and graphs0/2
Averages0/7
Mean deviation0/1
Correlation and regression0/3
Spread0/1
Variance and S.D.0/4
Comparing spread0/2

Ready for you

Up nextFrequency distributionsGroup data into classes; a class mark is the middle of a class, and cumulative frequency adds up the classes so farStart
The arithmetic meanStart $\bar{x} = \frac{\sum x_i}{n}$ ,

or

\frac{\sum f_ix_i}{\sum f_i}

with frequencies; deviations from the mean add up to

0

Range and its coefficientStartRange

= L - S

; coefficient of range

= \frac{L - S}{L + S}

; for classes, use the class limits

Variance of raw dataStart

\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2

Whole chapter

How the ideas build on each other. Each one opens once you've mastered what it needs.

Data and graphs0 / 2

Frequency distributionsGroup data into classes; a class mark is the middle of a class, and cumulative frequency adds up the classes so farStart
Histograms, ogives and pie chartsA histogram's areas show frequency; ogives plot cumulative frequency and cross at the median; a pie chart's angle is $\frac{f}{N} \times 360^\circ$ Locked

Averages0 / 7

The arithmetic mean $\bar{x} = \frac{\sum x_i}{n}$ ,

Mean deviation0 / 1

Mean deviationThe average distance from the mean (or median): $\frac{\sum |x_i - \bar{x}|}{n}$ ,

Correlation and regression0 / 3

The correlation coefficient $r = \frac{\operatorname{cov}(x, y)}{\sigma_x\sigma_y}$

Spread0 / 1

Range and its coefficientRange $= L - S$ ; coefficient of range $= \frac{L - S}{L + S}$ ;

Variance and S.D.0 / 4

Variance of raw data $\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n} = \frac{\sum x_i^2}{n} - \bar{x}^2$

Comparing spread0 / 2

Coefficient of variationC.V. $= \frac{\sigma}{\bar{x}} \times 100$ ;

or

\frac{\sum f_ix_i}{\sum f_i}

with frequencies; deviations from the mean add up to

0

Start

Shifting, scaling and correcting the meanThe mean of

a + bx_i

is

a + b\bar{x}

; a wrong or removed value changes the total

n\bar{x}

firstLocked

The combined meanTwo groups combine to

\frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}

: weight each mean by its group's sizeLocked

The mean of grouped dataUse class marks as the values; an assumed mean or step deviations keep the arithmetic smallLocked

The medianThe middle value of sorted data; with frequencies, find the class where the cumulative frequency passes

\frac{N}{2}

Locked

The mode and the empirical relationThe most frequent value; for moderately skewed data, mode

= 3

median

- 2

meanLocked

Geometric and harmonic meansG.M.

= \sqrt[n]{x_1x_2\cdots x_n}

; H.M.

= \frac{n}{\sum 1/x_i}

gives an average speed over equal distances; A.M.

\ge

G.M.

\ge

H.M.Locked

with frequencies as weights

Locked

lies in

[-1, 1]

;

shifting leaves it alone, and a negative scale flips its sign

Locked

Lines of regression

y - \bar{y} = b_{yx}(x - \bar{x})

with

b_{yx} = r\frac{\sigma_y}{\sigma_x}

; both lines pass through

(\bar{x}, \bar{y})

Locked

Working with two regression linesThey meet at the means;

r^2 = b_{yx}b_{xy}

, with

r

taking the coefficients' sign; tell the lines apart because

b_{yx}b_{xy} \le 1

Locked

for classes, use the class limits

Start

Variance with frequencies

\sigma^2 = \frac{\sum f_i x_i^2}{N} - \bar{x}^2

with

N = \sum f_i

; for classes,

x_i

is the class markLocked

Standard deviation

\sigma = \sqrt{\text{variance}}

, in the data's own units; it is never negativeLocked

Change of origin and scaleFor

u = \frac{x - a}{h}

:

\sigma_x = |h|\,\sigma_u

; adding a constant changes nothingLocked

the smaller C.V. is the more consistent

Locked

Standard deviation of combined data

\sigma^2 = \frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}

with

d_i = \bar{x}_i - \bar{x}

Locked