1. Very Basic Statistics

2. Course Content
Data Types
Descriptive Statistics
Data Displays

3. Data Types

4. Variables Quantitative Variable Qualitative Variable
A variable that is counted or measured on a numerical scale
Can be continuous or discrete (always a whole number).
Qualitative Variable
A non-numerical variable that can be classified into categories, but can’t be measured on a numerical scale.
Can be nominal or ordinal

5. Continuous Data Continuous data is measured on a scale.
The data can have almost any numeric value and can be recorded at many different points.
For example
Temperature (39.25oC)
Time (2.468 seconds)
Height (1.25m)
Weight (66.34kg)

6. Discrete Data Discrete data is based on counts, for example;
The number of cars parked in a car park
The number of patients seen by a dentist each day.
Only a finite number of values are possible e.g. a dentist could see 10, 11, 12 people but not people

7. Nominal Data
A Nominal scale is the most basic level of measurement. The variable is divided into categories and objects are ‘measured’ by assigning them to a category.
For example,
Colours of objects (red, yellow, blue, green)
Types of transport (plane, car, boat)
There is no order of magnitude to the categories i.e. blue is no more or less of a colour than red.

8. Ordinal Data
Ordinal data is categorical data, where the categories can be placed in a logical order of ascendance e.g.;
1 – 5 scoring scale, where 1 = poor and 5 = excellent
Strength of a curry (mild, medium, hot)
There is some measure of magnitude, a score of ‘5 – excellent’ is better than a score of ‘4 – good’.
But this says nothing about the degree of difference between the categories i.e. we cannot assume a customer who thinks a service is excellent is twice as happy as one who thinks the same service is good.

9. Task 1
Look at the following variables and decide if they are qualitative or quantitative, ordinal, nominal, discrete or continuous
Age
Year of birth
Sex
Height
Number of staff in a department
Time taken to get to work
Preferred strength of coffee
Company size

10. Descriptive Statistics

11. Session Content
Measures of Location
Measures of Dispersion

12. Measures of Location

13. Common Measures
Measures of location summarise the data with a single number
There are three common measures of location
Mean
Mode
Median
Quartiles are another measure

14. Mean
The mean (more precisely, the arithmetic mean) is commonly called the average
In formulas the mean is usually represented by read as ‘x-bar’.
The formula for calculating the mean from ‘n’ individual data-points is;
X bar equals the sum of the data divided by the number of data-points

15. Pro’s & Con’s Disadvantages Advantages
basic calculation is easily understood
all data values are used in the calculation
used in many statistical procedures.
Disadvantages
It may not be an actual ‘meaningful’ value, e.g. an average of 2.4 children per family.
Can be greatly affected by extreme values in a dataset. e.g. seven students take a test and receive the following scores.
The average score is 54.7 – but is this really representative of the group?
If the extreme value of 99 is dropped, the average falls to 47.3

16. Mode
The mode represents the most commonly occurring value within a dataset.
We usually find the mode by creating a frequency distribution in which we tally how often each value occurs.
If we find that every value occurs only once, the distribution has no mode.
If we find that two or more values are tied as the most common, the distribution has more than one mode.

17. Pro’s & Con’s Advantages Disadvantages easy to understand
not affected by outliers (extreme values)
can also be obtained for qualitative data
e.g. when looking at the frequency of colours of cars we may find that silver occurs most often
Disadvantages
not all sets of data have a modal value
some sets of data have more than one modal value
multiple modal values are often difficult to interpret

18. Task 2
The following values are the ages of students in their first year of a course
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18
Find the mean age of the students
Find the modal value
In your opinion which is the better measure of location for this data set?

19. Median
Median means middle, and the median is the middle of a set of data that has been put into rank order.
Specifically, it is the value that divides a set of data into two halves, with one half of the observations being larger than the median value, and one half smaller.
Half the data < 29
Half the data > 29

20. Finding the Median from Individual Data
Step 1:- Arrange the observations in increasing order i.e. rank order. The median will be the number that corresponds to the middle rank.
Step 2:- Find the middle rank with the following formula: Middle rank = ½*(n+1)
Step 3 – Identify the value of the median
If ‘n’ is an odd number the middle rank will fall on an observation. The median is then the value of that observation.

21. Finding the Median from Individual Data
If ‘n’ is an even number, the middle rank will fall between two observations. In this case the median is equal to the arithmetic mean of the values of the two observations
Position of Median = ½*(n+1) = 4.5
Median =

22. Pro’s & Con’s Advantages the concept is easy to understand
the median can be determined for any type of data (with the exception of nominal)
the median is not unduly influenced by extreme values in the dataset
Disadvantages
data must be arranged in rank order (ascending or descending)
cannot combine medians in statistical calculations as with mean values

23. Task 3 Using the student age data below, find the median age
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18

24. Quartiles Also known as percentiles
Lower quartile - 25% of the data is below this
Position of Q1 = ¼*(n+1)
Upper quartile – 75% of the data is below this
Position of Q3 = ¾*(n+1)
If a quartile falls on an observation, the value of the quartile is the value of that observation.
For example, if the position of a quartile is 20, its value is the value of the 20th observation.

25. Quartiles
If a quartile lies between observations, the value of the quartile is the value of the lower observation plus the specified fraction of the difference between the two observations.
Position of Upper Quartile = ¾*(n+1) = 6.75
Upper quartile = data-point *(data-point 7 – data-point 6)
Upper quartile = *(70 – 54) = 66

26. Task 4
Using the student age data below find the upper and lower quartiles
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18

27. Measures of Dispersion

28. Common Measures
The dispersion in a set of data is the variation among the set of data values.
It measures whether they are all close together, or more scattered. 4 2 6 8 10 12
Report turnaround time (days) 4 16 2 6 8 10 12 14
Report turnaround time (days)

29. Common Measures The four common measures of spread are the range
the inter-quartile range
the variance
the standard deviation

30. Range
The range is the difference between the largest and the smallest values in the dataset i.e. the maximum difference between data-points in the list.
It is sensitive to only the most extreme values in the list. The range of a list is 0 if and only if all the data-points in the list are equal. 4 16
Days
Range

31. Pro’s & Con’s Advantages best for symmetric data with no outliers
easy to compute and understand
good option for ordinal data
Disadvantages
doesn’t use all of the data, only the extremes
very much affected if the extremes are outliers
only shows maximum spread, does not show shape

32. Task 5 Using the student age data find the range of the data.
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18

33. Inter-quartile Range (upper quartile – lower quartile)
Essentially describes how much the middle 50% of your dataset varies
example: if all patients in a dentist surgery took more- or-less the same time to be treated with only one or two exceptionally quick or long appointments you would expect the inter-quartile range to be very small
but if all appointments were either very quick or very long, with few in between then the inter-quartile range would be larger.

34. Pro’s & Con’s Advantages Good for ordinal data Ignores extreme values
More stable than the range because it ignores outliers
Disadvantages
Harder to calculate and understand
Doesn’t use all the information (ignores half of the data-points, not just the outliers)
Tails almost always matter in data and these aren’t included
Outliers can also sometimes matter and again these aren’t included.

35. Task 6 Using the student age data find the inter- quartile range.
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18

36. Variance and Standard Deviation
(s2, s2) =(population notation, sample notation)
The variance (s2, s2) and standard deviation (s, s) are measures of the deviation or dispersion of observations (x) around the mean (m) of a distribution
Variance is an ‘average’ squared deviation from the mean

37. Variance and Standard Deviation
The standard deviation (SD) is the square root of the variance.
small SD = values cluster closely around the mean
large SD = values are scattered 8 12 10
1 SD
Mean 4 16
Mean 10
1 SD 6 8 12 14
Days

38. Variance and Standard Deviation
The following formulae define these measures
Population Sample

39. Variance Advantages: Disadvantages: uses all of the data values
the variance is measured in the original units squared
extreme values or outliers effect the variance considerably
hard to calculate manually

40. Standard Deviation Advantages: Disadvantages:
same units of measurement as the values
useful in theoretical work and statistical methods and inference
Disadvantages:
hard to calculate manually

41. Task 7
Using the student age data find the variance and the standard deviation
18, 19, 18, 25, 22, 20, 21, 45, 33, 20, 18, 18

42. Session Summary Measures of Location Measures of Dispersion Mean Mode
Median
Quartiles
Measures of Dispersion
Range
Interquartile Range
Variance
Standard Deviation

43. Data Displays

44. Session Content Histograms Run charts Box plots Bar charts
Pareto charts
Pie charts
Scatter plots
Contingency tables

45. Histograms

46. Run Charts

47. Boxplots

48. Bar Charts

49. Pareto Charts

50. Pie Charts

51. Scatterplots

52. Contingency Tables Colour of eyes Colour of hair Brown Green/grey Blue
Total
Black 50 54 41
145 38 46 48
132
Fair 22 30 31 83
Ginger 10 20 40
120
140
400=N

53. Session Summary Histograms Run charts Box plots Bar charts
Pareto charts
Pie charts
Scatter plots
Contingency tables

54. Course Summary
Data Types
Descriptive Statistics
Data Displays