Introduction to statistical concepts (population, sample, sampling, central tendency, spread). Mainly aimed at language teachers in advanced studies programmes (e.g., Masters courses)

1. INTRODUCTION TO STATISTICS FOR SECOND
LANGUAGE EDUCATORS
DR ACHILLEAS KOSTOULAS

2. OBJECTIVES OF THIS SESSION
 You will learn how to construct a sample
 You will learn how to describe your sample using statistical methods.
 You will learn how to find connections between different phenomena in your data.

3. OUTLINE OF THIS SESSION
1. Populations and samples
2. Different types of data
3. Univariate analysis
 Central tendency
 Spread
4. Bivariate analysis
 Cross-tabulations
 T-Tests
 Correlations

4. POPULATIONS & SAMPLES

5. POPULATION
 The total number of people (or events, or things) whose
properties or behaviour we are interested in
understanding
 e.g. University students in Austria
 Symbol: N
Population
Sampling frame
Sample

6. SAMPLING FRAME
 The total number of people (or events, or things) that we
have access to for our research
 e.g. Students currently present in this classroom
Population
Sampling frame
Sample

7. SAMPLE
 The total number of people who were contacted and
agreed to participate in the study
 Symbol: n
Population
Sampling frame
Sample

8. SAMPLING METHODS / STRATEGIES
 Simple random sampling
 Systematic sampling
 Convenience sampling

9. HOW LARGE SHOULD THE SAMPLE BE?
Depends on:
 the population size
 the degree of certainty
required
 the statistical tools we
want to use

10. DIFFERENT TYPES OF DATA

11. Variables
Cases
Values

12. LEVELS OF MEASUREMENT
1. Nominal data
2. Ordinal data
3. Scale data

13. CATEGORICAL / NOMINAL DATA
 A variable is nominal if values do not have a numerical relation to each other.
 Examples:
 Gender (M / F / Other)
 Place of Birth

14. ORDINAL DATA
 Ordinal variables are like categorical ones, but we can rank the values according to order, size,
frequency, etc.
 Examples:
 Level of education (High School, BA, MA/Mag., Doctorate)
 Attitudes (strongly disagree, disagree, neutral, agree, strongly agree)

15. CONTINUOUS / SCALE DATA
 A variable is continuous if it contains an infinite number of values that can be mathematically
manipulated.
 Examples:
 Age (12, 13, 13:2, 14…)
 Height (165cm, 167cm, 183cm…)

16. UNIVARIATE ANALYSIS
CENTRAL TENDENCY

17. MEASURES OF CENTRAL TENDENCY
 Mode (the most common value)
 Median (the middle value)
 Mean (the middle value, weighted)

18. EXAMPLE (RAW DATA)
Case Height Gender Loves Statistics
1 167 M Strongly disagree
2 178 M Strongly agree
3 189 F Agree
4 201 F Agree
5 182 M Disagree
6 175 F Strongly agree
7 162 M Strongly disagree
8 180 F Disagree
9 187 M Agree

19. EXAMPLE (PROCESSED)
Case Height Gender Loves Statistics
1 167 1 4
2 178 1 1
3 189 2 3
4 201 2 3
5 182 1 2
6 175 2 4
7 162 1 1
8 180 2 2
9 187 1 3
Total 1,621 - -

20. CENTRAL TENDENCY: THE MODE
Gender N %
--Male 5 55
--Female 4 44
--Total 9 100*
Case Gender
1 1
2 1
3 2
4 2
5 1
6 2
7 1
8 2
9 1
*Rounding up error
“The majority of respondents were male (n = 5, 55%). “
“Respondents were almost evenly split
between male (n = 5, 55%) and female (n = 4,
44%)”

21. CENTRAL TENDENCY: THE MEDIAN
Case <3
1 4
2 1
3 3
4 3
5 2
6 4
7 1
8 2
9 3
Case <3
1 4
6 4
3 3
4 3
9 3
5 2
8 2
2 1
7 1
I love statistics N %
--Strongly agree 2 22
--Agree 3 33
--Disagree 2 22
--Strongly disagree 2 22
--Total 9 100*
*Rounding up error
“As can be seen in Table 1, attitudes towards statistics
were largely positive (x̅ = 3)”

22. CENTRAL TENDENCY: THE MEAN
 1,441 / 9 = 180.1
Case Height
1 167
2 178
3 189
4 201
5 182
6 175
7 162
8 180
9 187
Total 1,441
“Respondents were rather tall (M = 180.1)”

23. UNIVARIATE ANALYSIS
SPREAD

24. COMPARE THESE TWO SCHOOLS
School A
School B
0
1
2
3
4
5
6
7
8
40-49 50-59 60-69 70-79 80-89 90-100
Based on Muijs 2007

25. COMPARE THESE TWO SCHOOLS
Case School A School B
1 45 60
2 50 65
3 55 65
4 60 70
5 65 70
6 70 70
7 70 70
8 75 70
9 80 70
10 85 75
11 90 75
12 95 80
Media
n
70 70
Mean 70 70

26. MEASURES OF SPREAD
 Range (the difference between the highest and the lowest value)
 Interquartile range (the difference between the highest and lowest values after we remove extremes)
 Standard deviation

27. MEASURES OF SPREAD: RANGE
Case School A School B
1 45 60
2 50 65
3 55 65
4 60 70
5 65 70
6 70 70
7 70 70
8 75 70
9 80 70
10 85 75
11 90 75
12 95 80
Media
n
70 70
Mean 70 70
Range
Range (School A): 95 – 45 = 50
Range (School B): 80 – 60 = 20
“The test scores in School A ranged from 45 to 95 (M
= 70. Scores in School B were more tightly
distributed, ranging from 60 to 80 (M = 70)”

28. MEASURES OF SPREAD: INTERQUARTILE RANGE
Case School A School B
1 45 60
2 50 65
3 55 65
4 60 70
5 65 70
6 70 70
7 70 70
8 75 70
9 80 70
10 85 75
11 90 75
12 95 80
IQR
IQR (School A): 82.5 – 57.5 = 25
IQR (School B): 72.5 – 67.5 = 5
“Although the average test performance in both
schools was similar (M = 70), the test scores in School
A were much more widely distributed than those in
School B (IQRA = 25, IQRB = 5”

29. MEASURES OF SPREAD: STANDARD DEVIATION
Case School A School B
1 45 60
2 50 65
3 55 65
4 60 70
5 65 70
6 70 70
7 70 70
8 75 70
9 80 70
10 85 75
11 90 75
12 95 80
Media
n
70 70
Mean 70 70
SD (School A) / SDA: 15.81
SD (School B) / SDB: 5.22
“The test scores in School A were satisfactory (M = 70,
SD= 15.81). School B reported similar results, which
clustered more tightly around the average (M = 70,
SD = 5.22)”

30. UNIVARIATE STATISTICS: SUMMARY
Central Tendency Spread
Mode Median Mean Range IQR SD
Nominal 
Ordinal    
Continuous    

31. POP QUIZ
 Average and mean are the same thing
In daily use, the words average and mean are interchangeable. In statistics, the mean is one type of average. The mode and
median are also types of average
 We must always use the median with ordinal variables
Technically, we can use both the median and the mode, but the median is a more powerful metric. The third option, the
mean, cannot be used with ordinal data.
 We must always use the median with continuous variables
It is usually the best option. However, if we have unusual data (with one or two very high or very low values) it may be better
to use the median.
 We can calculate mean values in a Likert scale (1: Strongly Agree; 2: Agree; 3: Disagree; 4: Strongly Disagree)
Some people do, but you shouldn’t. Likert scales produce ordinal data. You should not use the mean when your data is
ordinal.
 The appropriate spread metric for nominal variables is the IQR
No. Nominal data cannot be ranked in any sensible way, so they do not have a spread.

32. BIVARIATE ANALYSIS
CROSSTABULATIONS & CHI-SQUARE TESTS

33. CROSSTABULATIONS
 We use a cross-tabulation when we want to compare two ordinal or nominal variables
 Examples:
 Gender x Favourite colour
 School type x Attitudes towards mathematics

34. EXAMPLE CROSSTABULATION

35. CHI-SQUARE TEST
Action Figures Barbie Dolls
Male (50)
25 25
Female (60)
30 30

36. CHI-SQUARE TEST
Action Figures Barbie Dolls
Male (50) 48 2
25 25
Female (60) 25 35
30 30

37. CHI-SQUARE
“A statistically significant difference was found in the toy
preferences of boys and girls. As can be seen in Table 1
boys were much more likely to prefer action figures,
compared to girls (χ2= 36.068, df=1, p=o.ooo)“
Gender AF BD Total
--Male 48 2 50
--
Female
25 35 60
--Total 73 37 110

38. BIVARIATE ANALYSIS
T-TESTS

39. T-TESTS
 We use a t-test to see if there is any connection between a nominal variable (the independent variable)
and a continuous one (the dependent variable)
 The t-test breaks up your population in two groups (e.g., boys and girls), examines the mean value of the
independent variable for each group, and then compares them.

40. T-TESTS

41. T-TEST

42. BIVARIATE ANALYSIS
CORRELATIONS

43. CORRELATIONS
 We use a correlation (e.g., Spearmann or Pearson‘s coefficient) to see if there is any connection between
two continuous variables (e.g. weight and height).
 Correlations range from 1 to -1. A high value on either side means that the two variables are strongly
connected. A value close to 0 means that they are not.
 We can depict correlations visually with a scatterplot diagramme.

44. STRONG POSITIVE CORRELATION

45. STRONG NEGATIVE CORRELATION

46. NO CORRELATION

47. CORRELATION IS NOT CAUSATION!

48. REALLY, IT DOESN‘T

49. OR DOES IT?

50. SUMMARY
Nominal Ordinal Continuous
Nominal Crosstabs / χ2 Crosstabs / χ2 T-Test (if it has two
values)
Ordinal Crosstabs / χ2 Crosstabs / χ2 T-Test (if it has two
values)
Continuous T-Test (if it has two
values)
T-Test (if it has two
values)
Correlation

51. POP QUIZ
 If I want to test whether there is a connection music preferences and gender, I must use a cross-tab
That is correct. Music preferences and gender are both nominal variables. The correct procedure for pairing nominal
variables is a crosstab (and chi-square)
 A p value of 0.045 shows that something is statistically significant.
That is correct. The usual threshold of statistical significance in educational research is 0.05, and anything lower than
that is considered significant.
 I can prove that something is causing something else using a Pearson‘s correlation coefficient.
No, you cannot. Correlation does not imply causation.