Statistics Cheat Sheet

Topic Formula/Description Example Excel

Central Tendency:

Mean Sample:

Population: Given: 8, 5, 0, -7, -12
Mean = (8 + 5 + 0 + -7 + -12)/5
= -6/5 = -1.20

Median Middle value in ordered data.
If number of values (N) is odd:
Median = Given ordered data: -2, 0, 4, 7, 9
(N + 1)/2 = (5 + 1)/2 = 6/2 = 3
Median = (the value in position 3) = 4

If N is even:
Average the middle 2 values. Given ordered data: -8, -4, -2, 5, 7, 9
N/2 = 6/2 = 3 so average the 3rd and 4th values.
Median = (-2 + 5)/2 = 3/2 = 1.5

Mode The data value with more repeats than any other data value. Given: -4, 2, 0, 2, 3, 2, 3
Mode = 2 See WARNING below.

No Mode All data values have the same number of repeats. Given: -4, 0, 2, 3, 5
No Mode

Bimodal Two data values are tied for the most number of repeats. Given: -4, 6, 1, 1, 2, 6, 1, 5, 6
Bimodal values = 1 and 6 WARNING, Excel Mode formula gives
you only one of the two bimodal values.
Best to sort data and count repeats.

Multimodal More than two data values are tied for the most number of repeats. Given: -4, 0, -4, 0, 2, 3, 2, 5, 3
Multimodal values = -4, 0, 2 and 3 WARNING, Excel gives you only one
of the multimodal values.

Positive Skewness Most extreme data values are to the right side of centre.

Manual:

Excel: Skewness = 3(3 - 2) / 2.22 = 1.35 Mean: =AVERAGE(A1:A29) = 3
Median: =MEDIAN(A1:A29) = 2
Std.Dev: =STDEV(A1:A29) = 2.22
Skewness = 3*(3 - 2)/2.22 = 1.35
Using Excel to calculate with the "manual"
Skewness formula yields the same answer
as the manual calculation on the left.

Excel SKEW formula below has a different
answer. Ask your teacher which method
to use.
=SKEW(B121:B149)
= 0.736230686

Negative Skewness

Most extreme data values to left side of centre.

Skewness = 3(5 - 6) / 2.19 = -3 / 2.19 = -1.37

Mean: =AVERAGE(A1:A29) = 5
Median: =MEDIAN(A1:A29) = 6
Std.Dev: =STDEV(A1:A29) = 2.19
Skewness = 3*(5 - 6)/2.19 = -1.37

=SKEW(B121:B149)
= -0.983377

Variability:

Range Difference between largest & smallest values. Given: 3, 9, -1, 0
Range = Maximum - Minimum = 9 - (-1) = 10

Sample Variance

Data	x_i - x̄	(x_i - x̄)²

7	7 - 2 = 5	25
5	5 - 2 = 3	9
0	0 - 2 = -2	4
-4	-4 - 2 = -6	36
	Total:	74
s² = 74 / (4-1) = 74 / 3 = 24.6666667

Population Variance

Data	x_i - x̄	(x_i - x̄)²

7	7 - 2 = 5	25
5	5 - 2 = 3	9
0	0 - 2 = -2	4
-4	-4 - 2 = -6	36
	Total:	74
σ² = 74 / 4 = 74 / 4 = 18.50

Same as Above

Sample Standard Deviation Given: Variance = 9
s = square root of 9 = 3

Given data: 7, 5, 0, -4
Calculate variance: s² = 24.6666667
s = square root of 24.6666667 = 4.96655481 =stdev(A1:A4)

Population Standard Deviation Given data: 7, 5, 0, -4
Calculate variance: σ² = 18.50
σ = square root of 18.50 = 4.30116263 =stdevp(A1:A4)

Sample Coefficient of Variation Given: mean = 80, s = 10
Sample CV = 10/80 X 100% = 12.5%

Population Coefficient of Variation Given: mean = 75, σ = 5
Sample CV = 5/75 X 100% = 6.67%

Interquartile Range A Variability measure, but described below, under "Position".

Empirical Rule 68% within 1 standard deviations of mean
95% within 2 standard deviations of mean
99.7% within 3 std. deviations of mean

Position:

Percentile Where i is position in ordered data of "percentile" value.
If i is a decimal: round up
If i is a whole number: average i and i+1 Find the 25th percentile, given: 1, 3, 4, 4, 5
i = (25/100)5 = .25 x 5 = 1.25
i is a decimal so round up to 2
The value in position 2 is 3 so the 25th percentile is 3.

Find the 50th percentile, given: -8, -5, 3, 9
i = (50/100)4 = 2 which is a whole number, so average the values at positions 2 & 3:
(-5 + 3)/2 = -2/2 = -1
-1 is the 50th percentile. Use Excel percentile function only if your teacher agrees.

Interquartile Range Q₃ - Q₁ Given sorted data: 2, 3, 4, 5, 6, 7, 8, 9, 10
Q3: i = 75/100(9) = 6.75 (round up to 7)
Data value at position 7 is 8
Q1: i = 25/100(9) = 2.25 (round up to 3)
Data value at position 3 is 4
Interquartile Range = Q3 - Q1 = 8 - 4 = 4

Sample Z Score Find z score for 50.
Given: mean = 100, standard deviation = 10
z score = (50 - 100) / 10 = -5 =(A1-A2)/A3 where:
A1 = some data value
A2 = mean
A3 = standard deviation

Population Z Score Find z score for 150.
Given: mean = 100, standard deviation = 25
z score = (150 - 100) / 25 = +2 Same as above.

Probability:

Permutation Pick 2 students out of 5; give one $5 & the other $500.
You choose permutation because order is important: $5 to Lee & $500 to Joe is different than $5 to Joe & $500 to Lee.

Combination If you could choose 2 of the following 5 prizes: Roadster, Highlander Hybrid, MKS, GLK or cash. How many different pairs could be selected?
Order doesn't matter. (cash, MKS) is the same as (MKS, cash) to you, so use Combination. =COMBIN(5,2)
= 10

How many ways could you can group 2 dogs out of collection of 4 dogs?
(Lassie,Rex) is the same as (Rex,Lassie), so use Combination: =COMBIN(4,2)
= 6

Discrete Probability:

Expected Value Given: x P(x)
0 .13
1 .33
2 .54

Binomial Mean Given: N = 20, p = .30

Discrete Variance

Given: X	Probability
0	.65
1	.10
2	.20
3	.05

= 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65

= (0-.65)².65 + (1-.65)².1 + (2-.65)².2 + (3-.65)².05
= .4225(.65) + .1225(.10) + 1.8225(.20) + 5.5225(.05)
= .274625 + .01225 + .3645 + .276125
= .9275

In the Excel images above, formula comments are shown to the left of each cell that contains a formula.

For a binomial distribution, with probability of .5 and sample size of 10, what is the variance?
Variance = σ² = np(1-p) = np(1-p) = 10(.5)(1 - .5) = 5(.5) = 2.5

Discrete Standard Deviation

Given: X	Probability
0	.65
1	.10
2	.20
3	.05

= 0(.65) + 1(.10) + 2(.20) + 3(.05) = .65

= (0-.65)².65 + (1-.65)².1 + (2-.65)².2 + (3-.65)².05
= .4225(.65) + .1225(.10) + 1.8225(.20) + 5.5225(.05)
= .9275
Take the square root of the variance:
Standard deviation = σ = √.9275 = .963068

In the Excel image above, formula comments are shown to the left of each cell that contains a formula.

Binomial Standard Deviation For a binomial distribution, with probability of .5 and sample size of 10, what is the standard deviation?
Variance = σ² = np(1-p) = np(1-p) = 10(.5)(1 - .5) = 5(.5) = 2.5
Standard deviation = σ = √2.5 = 1.58113883

Formula Comparisons (for population data)

	Univariate	Discrete	Binomial
`Mean`
`Variance`
`Standard Deviation`