1. Statistics -S1

2. Chapter 1 Mathematical models in probabilities and statistics

3. Mathematical models A simplification of a real world situation
Adv.: quick and easy to produce, can simplify a more complex situation, enables for predictions to be made and can help to provide control
Disadv.: only give a partial description of real situation and they only work for a certain range of values

4. Chapter 2 Representation and summary of data- location

5. Quantitative variables
Variables associated a numerical value (e.g. height)

6. Qualitative variable
Variables which do not have a numerical value (e.g. hair colour)

7. Continuous variable
A variable that can take any value in a given range

8. Discrete variable
A variable that can only take integral values within a given range

9. Mode/ modal class The value or class that appears most often
E.g. 1, 2, 2, 2, 3, 3 ,4, 5 mode= 2

10. Mean x = ∑ x n Where… n = no. of observations
∑ x = sum of observations
x = mean of the sample

11. Mean of a combined set of data
x = n1x1 + n2x2
n1 + n2
Where…
x = mean
n = size of the sample
xn = mean of individual sample

12. Frequency distribution table - mean
x = ∑ fx
∑ f
Where…
∑ fx = frequency multiplied by class or midpoint
∑ f = sum of the frequencies
x = mean of the sample

13. Median The middle value of ordered data
To find the position where the median lies… n 2
If the position isn’t an integer, round up
Where n= number of observations

14. Interpolation Length of pine cone (mm) No. of pine cones, f
Cumulative frequency
30-31 2
32-33 25 27
34-36 30 57
37-39 13 70
Median= 70/2 =35th value
33.5 Q2
36.5
Q2 – =
36.5 – 27 35 57
Make Q2 the subject
Q2 = 34.3

15. Coding Used to make large values easier to work with
General form : y = x – a b
No effect on product moment correlation coefficient
Coded regression line may not be the same as the actual line

16. Chapter 3 Representation and summary of data-dispersion

17. Range
Highest value - lowest value

18. Lower quartile, Q1 n 4
If n isn’t an integer, round up to find the corresponding position
Where n = sample size

19. Upper quartile, Q3 3n 4
If n isn’t an integer, round up to find the corresponding position
Where n = sample size

20. Interquartile range Q3 - Q1 Where… Q1 = lower quartile
Q3 = upper quartile

21. Percentiles Split the data into 100 parts xth percentile = xn 100
Where n = sample size

22. Variance Represents the spread of a set of data = (∑ x )2 - ∑x 2 or =
n n fx fx
Remember: “Mean of the squares minus square of the mean

23. Standard deviation, σ
√variance

24. Chapter 4 Representation of data

25. Stem and leaf diagrams

26. Back-to-back stem and leaf diagrams
Used to compare two sets of data

27. Outliers Extreme values within the data
Plot outliers on boxplots with an x
Extreme values within the data
Outlier above upper quartile, Q3:
Q3 + (1.5 x interquartile range)
Outlier below lower quartile, Q1 :
Q1 - (1.5 x interquartile range)

28. Box plots Highest value Lowest value Upper quartile Lower quartile
Median

29. 3(mean – median) standard deviation
Skewness
3(mean – median) standard deviation
+ive number  + skew
-ive number  -ive skew
Close to 0  symmetrical

30. Positive skew
Mode < median < mean
Q2-Q1 < Q3-Q2

31. Negative skew
Mode > median > mean
Q2-Q1 > Q3-Q2

32. Symmetrical
Mode = median = mean
Q2-Q1 = Q3-Q2

33. Histograms Shows data distribution Continuous data
No gaps between bars
Area of bar α frequency

34. Frequency density
Frequency density = frequency class width

35. Area = k x frequency

36. Chapter 5 Probability

37. Venn diagrams
Whole rectangle represents sample space. Total probability = 1
Closed curves represent the outcomes for each event

38. P(A)

39. P(A’)

40. P(B)

41. P(B’)

42. P(A n B)

43. P(A u B)

44. P(A’ n B’) = P(A U B)

45. P(A’ U B’) = P(A n B)

46. P( A’ n B)

47. P( A n B’)

48. P(event A or event B or both)
P(A U B)

49. Complementary probability
P(A’) = 1 – P(A)

50. Addition rule
P(AUB)= P(A) + P(B) – P(A B)

51. Conditional probability
P(A given B) = P( A|B) = P(A B)
P(B)

52. Multiplication rule
P(A B) = P(A|B) x P(B)
P(A B) = P(B|A) X P(A)

53. Independent
P(A B) = P(A) X P(B)
P(A|B) = P(A)
P(B|A) = P(B)

54. Mutually exclusive
P(A B) = O

55. Chapter 6 Correlation

56. Positive correlation Most points lie in 1st and 3rd quadrants
Product moment coefficient correlation is closer to 1

57. Negative correlation Most points lie in the 2nd and 4th quadrants
Product moment correlation coefficient is closer to -1

58. No correlation Points lie in all four quadrants
Product moment correlation coefficient is O

59. Product moment correlation coefficient, r
A measure of linear relationship
r = Sxy
√SxxSyy
Where…
Sxy = ∑xy - (∑x ∑y) n
Sxx = ∑x2 - (∑x)2
Syy = ∑y2 - (∑y)2

60. Chapter 7 Regression

61. Independent (explanatory variable)
The variable that is set independently of the other variable
Plotted on the x-axis

62. Dependent (response) variable
The variable whose values are determined by the values of the independent variable
Plotted on the y-axis

63. Equation of regression line
y = a + bx
Gradient. For every increase in x, y increases by a factor of the gradient
Y-intercept. When x is zero, y is equal to the value of a
Where…
b= Sxy a = y - bx
Sxx

64. When to use the regression line
When the points form/almost form a straight line

65. Coding in regression lines
To turn the coded regression line into the actual regression line, substitute the codes into the answer

66. Interpolation
When a value of the dependent variable is estimated within the range of the data

67. Extrapolation
When a value is estimated outside of the range of the data
Unreliable

68. Chapter 8 Discrete random variables

69. Variable Represented by X, Y, A, B etc..
Can take on any specified set of values

70. Random variable
The value of a variable that is an outcome of an experiment, e.g. Rolling a die
Discrete  only on a discrete scale
Continuous  outcome can be any value on a continuous scale

71. Sample space The list of all possible outcomes of an experiment
E.g. Spinning a four-sided and a three sided spinner at the same time:

72. Probability distribution
A table showing the probability of each outcome in an experiment X 1 2 3 4 5 6
P(X=x)
1/6
Remember: All of the probabilities add up to one for discrete random variables

73. Cumulative distribution function, F(x)
Shows the running totals of the probabilities X 1 2 3 4 5 6
P(X=x)
1/6
F(x)
2/6
3/6
4/6
5/6
6/6

74. Expected value, E(X)
The total of the x values multiplied with the corresponding probabilities , ∑xP(X=x)
E.g. (1 x 1/6) + (2 x 1/6) + (3 x 1/6) + (4 x 1/6)+ (5 x 1/6) + (6 x 1/6) = 3.5 X 1 2 3 4 5 6
P(X=x)
1/6

75. E(X2)
Square the x values, multiply with their corresponding probabilities then total
E.g. (12 x 1/6) + (22 x 1/6) + (32 x 1/6) (42 x 1/6)+ (52 x 1/6) + (62 x 1/6) = 91/6 X 1 2 3 4 5 6
P(X=x)
1/6

76. Variance of a random variable
Var(X) = E(X2) – (E(X))2

77. E(aX+b)
E(aX+b) = aE(X) + b

78. Var(aX+b)
Var(aX+b) = a2Var(X)

79. Mean using coded data E.g. Y = X – 150 Mean of coded data = 5.1 50
Step 1 : rearrange making X the subject X= 50Y +150
Step 2 : Make E(X) the subject and solve E(X) = E(50Y +150)
=50E(Y) +150 =
=406

80. Standard deviation of coded data
E.g. Y = X – 150 σ = 2.5 50
Step 1 : Var(X) = Var( 50Y +150)
=502Var(Y)
= 502 x 2.52
= 15625
Step 2 : Standard deviation = √15625
= 125

81. Discrete uniform distribution
Probabilities are the same (e.g. rolling a die)
E(X) = n + 1 2
Var(X) = (n+1)(n-1) 12

82. Chapter 9 The Normal Distribution

83. Standard normal variable, Z
Z ~ N(0, 12)
Normal
Standard deviation, σ2, is 1
“is distributed”
Mean,μ , is 0

84. Normal distribution curvex
f(x) μ α
Area under curve represents probability. Total = 1
P(α<x)

85. Standardised curve z f(z) μ= 0 α
Area under curve represents probability. Total = 1
P(α<z)

86. P(Z < α) Step 1 – Draw curve
Step 2 - Find the probability in the table T
Step 3 - look at the corresponding z value to find the value of α z α

87. P(Z > α) Step 1 – Draw curve Step 2 - Find P(Z < α)
Step 3 – Subtract the answer from 1 z α

88. Random variable, X
X ~ N (μ, σ2)

89. Finding z from Random variable, X
If you are given a random variable, X, (e.g. 180kg) rather than z, find z-value using…
Z = X – μ σ

90. Random variable example
Find P(X < 53) given the random variable X ~ N(50, 42)… Step 1 – Sub-in values: P z < = Step 2 – Find probability using table. P(Z<0.75) =

91. Simultaneous equations to find σ and μ
E.g. P(X>35) = 0.05, P(X<15) =
Step 1- Draw curve
Step 2 – Look at table to find z values for each
Step 3 – Sub into Z = X – μ for each value σ
Step 4- Use substitution method to obtain μ and σ

92. Probability between two values
E.g. P(168 < z < 174), σ = 3.5, μ= 165
Step 1 – Draw curve
Step 2 – Sub in values into and solve Z = X – μ σ
This obtains P(o.86 < z< 2.55) for this example
P(Z<2.55) – P(Z<0.86) = – =