Helia Formulas 7 (7)

HELIA FORMULAS 7 (7)

BITE / Applied Mathematics 12.2.2008

Statistics

Median from grouped data:

LMd = lower class boundary of median class

n = total number of observations

FMd-1 = cumulative frequency of the class preceding median class

fMd = frequency of median class

cMd = width of median class

Quartiles (Q1 ja Q3), deciles (Dn) and percentiles (Pn%) can be defined by applying the same principle.

Arithmetic mean (=average) from ungrouped and grouped data:

xi = single observation fi = frequency of class i

n = total number of observations xi = midpoint of class i

n = total number of observations

Range and Range-Width (R):

[min ; max] R = max - min

Standard deviation from ungrouped data (with and without arithmetic mean):

From grouped data:

Variance (s2) and proportional variance (V):

s2 = s2 (square of standard deviation)

Linear regression:

y = a + bx

Pearson’s coefficient of correlation (rxy):

Spearman’s coefficient of rank correlation (rs):

di = difference between ranks of i

n = total number of ranked items

Probability

Theoretical probability

k = number of ways as event can occur

n = total number of different equally likely events that can occur

P(-A) = 1 – P(A)

-A = “Not A” = complement of A

P(A and B) = P(A) × P(B|A)

P(B|A) = P(B) when A does occur

Independent events, A ^ B :

(1) P(A and B) = P(A) × P(B)

P(A or B) = P(A) + P(B) – P(A and B) Þ

(2) P(A or B) = P(A) + P(B) - P(A) × P(B)

Mutually exclusive events: [P(A and B) =0]

P(A or B) = P(A) + P(B)

Permutations and combinations:

Factorial notation:

n! = n× (n-1) × (n-2) × … × 3 × 2 × 1

0! = 1

In how many different ways can one put n items in order? n!

Permutation:

If there are n items and k are to be placed in order, the number of different ways in which this can be done is:

Combination:

If there are n items and k are to be placed irrespective of order, the number of different ways in which this can be done is:

Probability distributions:

Binomial distribution:

p = the probability of ‘success’

q = 1 - p = the probability of ’failure’

n=number of trials

k= number of ‘successes’

Poisson distribution:

m = mean number of successes [= np]

n=number of trials

k= number of ‘successes’

e=exponential constant = 2,71828

Exponential distribution: P(>T) = e-aT P(<T) = 1 - e-aT

T = waiting time for the next ‘failure’

a = constant failure rate (= failures per time unit)

e=exponential constant = 2,71828

Normal distribution:

· Standard normal distribution:

· See z statistic in full on the separate sheet

If x ~N(m,s), then ~ N(0,1)

Normal estimate of binomial and Poisson distributions:

Estimate of populations mean:

m = population mean

= sample mean

cr = critical factor based on the significance level (see below)

s = sample standard deviation

n = total number of observations in sample

level of accuracy / 95 % / 99 % / 99,9 %
factor (cr) / 1,96 / 2,58 / 3,30

Estimation of the percentage value in population:

p = percentage value in population

= percentage value in sample (in decimal format)

= 1 -

n = total number of observations in sample

cr = critical factor based on the significance level (see the previous page)

Statistical testing

Critical values of the z-test:

5 % / 1 % / 0,1 %
two-tailed test / 1,960 / 2,576 / 3,291
one-tailed test / 1,645 / 2,326 / 3,090

Critical values of the t-test:

· See the separate sheet. Degrees of freedom f = n – 1
n = total number of observations in sample

Mean test

Test variable:

x = sample mean

m = comparison mean

s = population (or sample) standard deviation

n = total number of observations in sample

z -test, if n ³ 30

t -test, if n < 30

Percentage value test (only if n > 30)

tai

x = number of ’successes’ in sample

n = total number of observations in sample

p0 = comparison percentage value

q0 = 1 - p0

P = x/n : percentage value of ‘successes’ in sample (in decimal format)

Critical values of z-test: see above.

Two sample mean test (t -test):

where

x1, s1, n1, : mean, standard deviation and number of observations in sample 1

x2, s2, n2, : mean, standard deviation and number of observations in sample 2

Degrees of freedom: f = n1 + n2 -2

Two sample percentage value test (z -test):

where

P1 = percentage value of ‘successes’ in sample 1 (in decimal format)

P2 = percentage value of ‘successes’ in sample 2 (in decimal format)

n1 = number of observations in sample 1

n2 = number of observations in sample 2

c2 –test (Chi squared –test)

Critical values: see the separate sheet

Distribution test:

oi = observed values

ei = expected values :

Degrees of freedom: f = number of categories (usually = number of columns) - 1

Independency test:

oij = observed values

eij = expected values :

Degrees of freedom: f = (nr. of rows -1) × (nr. of columns -1)

Queuing theory

l = average number of clients joining the queue within a time unit T

m = average number of clients that could be served within a time unit T

Time unit T can be chosen freely.

’Jam factor’ : r =

Single line system M/M/1/¥/FIFO defaults:

· (1) Number of customers in the system:

· (2) Number of customers in queue:

· (3) Time spent in the system:

· (4) Queuing time:

Time from formulas (3) and (4) is presented in the same unit to which parameters l and m refer.