INTRODUCTION:
UNIT 12: STATISTICAL ANALYSIS OF EDUCATIONAL INDICATORS Objectives: To introduce learners to a range of basic statistical techniques and how
they can be used to analyse relationships among educational indicators.
On successful completion of the unit participants will be able to: 1. Define statistics
2. Explain the difference between Rate and Ratio
3. Identify and explain some of the basic statistical techniques and how they can be obtained using SPSS.
4. Use basic statistics to analyse the relationships among educational indicators 5. Appreciate the use of SPSS as a computerised statistical analytical tool
What is Statistics?
Statistics consist of a set of methods and rules for organising, analysing and interpreting data. There are two basic goals of statistics: summarisation and inference. Summarisation is the process of reducing large volumes of data into a few quantities that adequately represent or describe the data that they can easily be understood. The resulting values for doing this are known as descriptive statistics. Examples are Mode, Median, Mean and Standard deviation.
Inferential statistics on the other hand concerns the process of making statements about the population from your sample and encompasses a variety of procedures to ensure that the inferences are sound and rational, even though they may not always be correct. Most of the major inferential statistics come from a general family of statistical models known as the General Linear Model. This includes the t-test, Analysis of Variance (ANOVA), Analysis of Covariance (ANCOVA), regression analysis and so on. What is data?
Data is a collection of facts, such as values or measurements. Data can be qualitative or quantitative. Qualitative data is descriptive information (e.g. the school has nice surroundings or clean toilets). Quantitative data is numerical information (e.g. the school has 20 teachers or pupil-teacher ratio is 50 to 1).
What are the differences between Ratio and Rate
A ratio is a comparison of two numbers and can be written multiple ways (like 1/30 or 1:30). You typically do not use units, but if you do, they are often the same. If there are 1 teacher and say 30 pupils in a class, then we can say that the pupil-teacher ratio (PTR) is 30:1.
A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h. For rates, your units are different and often in distance and time. Another rate could be some other number per unit time, such as 100kb/sec. In this case, you could say "for every one second, my computer can download 100kb of data." In three seconds, your computer would download 300kb given this rate.
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In terms of educational indicators a rate could be some other educational characteristics in the population expressed as a percentage of the total population. E.g. the Gross Admission Rate (GAR) is defined as the Total number of new entrants in the first grade of primary education regardless of age, expressed as a percentage of the population at the official primary school-entrance age. So for example GAR of 80% for a country means that the capacity of the educational system in that country to provide access to Grade 1 for the official school-entrance age population is running at 80%. What are Education Indicators?
Education indicators can be regarded as descriptive statistics that serve as yardsticks that can tell us how well the education system is functioning. Examples of these indicators are:
Pupil Teacher Ratio (PTR): This indicator measures the number of teachers in relation to the size of the pupil population. This is useful for comparison against a nationally established norm.
Gross Admission Ratio (GAR): This indicator tells us about the general level of access to primary education in a given area. It also indicates the capacity of the education system to provide access to Grade 1 for the official school entrance age population. Measures of central tendency
Values describing central tendencies are useful for summarising an entire data by a single score that best represents the level of distribution when reporting research findings. There are three common measures of central tendency, the Mode, the Median and the Mean, each appropriate for a different level of measurement.
a) The Mode
The mode is the simplest measure of central tendency. It is simply the value of the observation that occurs most frequently in the distribution. The mode or modal score in the following example is 24 since 4 people obtained that score in the set. With large numbers of observations there may be more than one mode. This is called a bimodal distribution if there are two modal values or multimodal if there are more.
Example: 23 28 20 24 9 24 24 21 18 19 24 (b) The Median
The word median means ‘middle item’ It is used primarily for ordinal variables but also appropriate for interval/ratio variables. It can be found by rearranging a series of scores in a descending order of magnitude. The median is the central value after the scores have been arranged in a rank order. This means that there are as many scores or cases above the median as below it. The median identifies the position of an observation. Consider the test scores of nine students in the Example below.
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Example
16 6 11 24 17 4 19 9 20
Rearranged in descending order of magnitude these scores become 1st 2nd 3rd 4th 5th 6th 7th 8th 9th
24 20 19 17 16 11 9 6 4
median
When the number of series of scores is even the median value will lay half way between the 5th and 6th observations as in this Example .
16 29 20 9 34 10 23 12 15 22 Arranged in order of magnitude these scores become
34 29 23 22 20 16 15 12 10 9 = 20 + 16 divided by 2 Median = 18
(c) The Arithmetic Mean
To the lay person, the mean is the ‘average’, calculated by adding all of the scores together and then dividing by the number of scores. In statistical language, the figure obtained by carrying out this procedure is referred to as the Arithmetic Mean. The mean (M) is simply the sum ∑ (this Greek letter stands for ‘the sum of’) of all the scores X divided by the number (N) of scores.
M = ∑X N
Consider five people with ages on an interval scale: 19, 25, 20, 21, 17 (years) 102
By the above formula the mean M = 19 + 25 + 20 + 21 + 17 = 20.4 5
The Mean is the only measure that reflects the influence of all scores in the distribution and as such is adversely affected by extreme values whereas the Median is resistant to such values.
Consider the monthly incomes of 10 parents from a neighbourhood school: 200 200 200 200 250 250 250 260 260 20,000 Ghana Cedis
The median income for this sample is GH¢250. The mean income however, is Gh¢2207.
Which of these values will a head teacher, for example report? This may depend on intentions. If the head teacher is begging the Ghana Education Service for money for the school, then he or she will be a fool not to use the Median income of the parents to support his or her efforts. If however, he or she wants to attract reasonably ‘well off’ parents to send their children to the school, believing that this would enhance the effectiveness of the PTA, then perhaps using the mean income in adverts might do the trick.
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Measures of Dispersion or Spread
So far we have shown that it is possible to describe variables (factors that can be measured) in terms of their measure of central tendency (mode, median or mean). Another important feature of distribution, which is more appropriate for continuous (e.g. % academic performance) variables, is their spread or dispersion. Let us assume that you gave your students a maths exam, and to your surprise, every one of them obtained say 100%. In this case, this variable, Maths Performance, would have no spread. So, as far as your students‘ maths performance scores are concerned, there would be no variation. In other words, one student‘s score would not differ from another‘s. Therefore, there would be no point for example, in finding out if the girls did better than the boys. In fact, you would not be able to make any comparisons within this student group as far as their maths performance is concerned. However, in reality, we all know that this situation is highly unlikely to occur because there is usually wide variation between students‘ academic performance. This variation is what makes educational research interesting, because most of our efforts are geared toward finding out just what makes one student perform better academically than another in a seemingly similar academic environment. For these and other reasons we need to find ways of describing the spread of scores caused by the variation. This would allow us to measure the spread, compare different subpopulations (e.g. males and females) and possibly explain the variation.
Let us start with the simplest way of describing variation, the range, and then proceed to more complex measures, the variance and standard deviation.
(a) The Range
The range is simply the difference between the highest and the lowest score and therefore relies on only two values. Consider the following scores:
2, 5, 6, 7, 9, 99 range = 99 – 2 = 97 2, 5, 6, 7, 9, 11 range = 11 – 2 = 9
Clearly, the score of 99 in the first set of scores is at odds with the rest of the scores in the set and the range of 97 gives a false impression of the actual scores. Furthermore, the range can depend on the size of the sample. Usually, one would be expected to get a larger range for bigger sample sizes where you have a greater chance of getting extreme scores.
One solution to this problem of relying on just two values is to calculate what is known as the interquartile range (IQR). This is simply the value for the lower quartile subtracted from the value of the upper quartile which you can obtain the lower and upper quartile values from SPSS. I will show you how this is done in the practice sessions that follow or refer to Ofori & Dampson, 2011).
(b) The Variance
The variance is a measure of the average of the squared deviations of individual scores from the mean. It is based on the deviations or distances of each observation or score from the mean score. It is therefore the average error between the mean and individual scores or observations. The problem with the variance as a measure is that it gives us a measure in units squared, which does not make sense in the real world when you want to describe the spread of scores. For example, by saying that the average error (variance) in the students‘ exam performance in a subject is 2.2 marks
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squared sounds odd indeed. Nevertheless, it is an important measure and you will come across it a great deal in statistics. I am going to spare readers of the long steps, equations and formulae needed to calculate the variance (I can hear a sigh of relief) since by the click of the mouse, SPSS will give you the variance of your data (refer to Ofori & Dampson (2011) for step by step instructions. Readers who are interested in the mathematical journey of how the variance is arrived at should consult other statistics books, particularly, the introductory ones (see for example, Field, 2005; Fielding & Gilbert, 2000).
(c) Standard Deviation.
For the reason given above that it sounds odd to describe the spread of data by using the variance, we often take the square root of the variance which ensures that the measure of average error is in the same units as the original. This measure is known as the standard deviation (SD) and is simply the square root of the variance. The following are some of the features of SD
It is an important measure of dispersal because it measures how well the mean represents the data.
It reflects the amount of spread that the scores exhibit around the mean. The larger the SD the more spread out are the data. Conversely, the smaller
the SD the less spread out and the more similar are the data.
An SD of zero occurs when all scores are the same so there is no deviation around the mean.
Without given the SD, one cannot use only the mean to make an accurate assessment regarding the spread of data.
We are not going to worry you with the mathematics of arriving at the variance, because good old SPSS can do that for you. SPSS will give you the SD of your data through the Analyse – Descriptive Statistics – Frequencies procedure (see Ofori & Dampson 2011).
Frequency distributions
One way of exploring data set is to count the frequency (i.e. the number of times) that certain things happen. This provides you with frequency distributions, enabling you to compare information between groups of individuals. Such information can be displayed using histograms, bar charts or pie charts. They are also useful for describing your sample characteristics, such as age and sex, to demonstrate to readers that perhaps your sample is not biased.
The Normal Curve
Frequencies are also distributions and statisticians have calculated the probability of certain scores occurring in a normal distribution with a mean of 0 and a standard deviation of 1 (i.e. 68.26% of scores will lie between -1 and +1 SD of the mean, 95.44% between +-2 sd and so on, see Fig.12.1. On a graph (see Figure 12.1) this is known as the normal curve. The concept of the Normal curve help us to assess the normality of distribution of data (i.e. skewness and Kurtosis), standardise scores with different measurement for comparison and it is used as the basis for making statistical inferences. It has been shown that data which are influenced by many independent random effects have an approximately normal distribution. Scientific measurements,
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Educational indicators, i.e. Public Expenditure on Education as a percentage Total of Government Expenditure (%PXE) or Pupil-Teacher Ratio (PTR) better predicts pupil Academic performance, we will perform multiple regression.
(c) T-test
At times your question might be in looking at the difference between two variables rather than the correlation between them. Let us say, you wondered whether the girls were doing better than boys in mathematics in the district and you want to test whether this is true. To test for the difference between the two groups on mathematics performance you will perform a t-test which in simple terms, is a statistical analysis that compares the mean performance scores between the two groups. The application exercises provided at the end of this unit will give more understanding.
(d) Time Series Analysis
A time series is a collection of observations of well-defined data items obtained through repeated measurements over time. For example, measuring the expenditure of a school on regular items such as, infrastructure, office stationery, T & L materials, transport, etc each month of the year would comprise a time series. This means that data on expenditure that is well defined and is consistently made by the school at equally spaced intervals can be analysed using time series technique. Data collected irregularly or only once are not time series.
The basic objective of using time series is to analyse the trend (long term direction), the seasonal (systematic, calendar related movements) and the irregular
(unsystematic, short term fluctuations) in for example, school expenditure.
Application Exercise 1 (Distributions)
The following exercise will provide you with the opportunity of working with some basic statistics using SPSS. Please input the data below into SPSS
Table 1 GDP and GER data
Using the data in Table 1
1. Use SPSS to find the Maximum, Minimum, Mode, Median Mean, SD, Skewness and Kurtosis of GDP and GER
Years GDP per capita (GHC) Gross Enrolment rate (GER) 2000 300 25 2001 560 55 2002 605 61 2003 780 55 2004 800 55 2005 900 70 2006 900 80 2007 1000 120 2008 1200 120 2009 1500 140 2010 1600 150
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2. What do the Mean and SD of GDP and GER tell you about their consistency over the years?
3. What do the skewness of GDP and GER tell us about their normality of distribution using graphs?
Application Exercise 2 (Frequencies and graphical displays)
Suppose that we have collected the data in Table 2, use SPSS to obtain, bar or pie charts to answer the following questions:
How many or what % of students obtained first or upper second class? Analyse the percentages of degree class according to gender.
Table 2 Record of students‘ performance across class of degree
Application Exercise 3 (Correlations)
Going back to Table 1 above:
1. What is the correlation between GDP and GER?
2. What is the interpretation of the relationship between GDP and GER? 3. What may be the probable reasons for the relationship?
Using Table 3 below:
1. Interpret the correlation matrix of educational indicators and discuss the relationship between them.
Specifically,
2. Is the correlation between PTR and r statistically significant?
3. What is the value of the correlation coefficient between them and what does it mean?
4. What is the direction of the correlation between them and what does it mean? 5. Describe the result of the correlation between them
6. Based on the result what can you say about the relationship between them? Degree Class Count Male Female
first 4 3 1 upper-second 36 16 20 lower-second 48 32 16 third 5 4 1 pass 6 4 2 fail 1 1 0 Total 100 60 40
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Table 3. SPSS output of correlation Matrix (Pearson r) of Educational indicators
*Correlation is significant at p < 0.05 **Correlation is significant at p <0.01
GPI = Gender Parity Index; PTR = Pupil Teacher Ratio; %FT = Percentage of Female Teachers; GER = Gross Enrolment Ratio; AIR = Apparent Intake Rate; r = repetition rate; TR = Transition rate
7. In school A, the correlation (r) between the number of students per teacher and mean BECE performance is r = -0.50. In school B, the r = 0.1. If we assume that the relationships are causal, which of the following actions make sense? A. Move teachers from school A to school B
B. Move teachers from school B to school A C. Reduce class size in both schools
Application Exercise 4 (T-test)
The following dataset in Table 4 on mean % BECE performance in core subjects was obtained from some Schools in the district. Input the data set in into SPSS. When assigning values in SPSS, give a code of 1 for boys and 2 for girls.
Questions:
1. Which of the sexes was more consistent in performance?
2. Which analytical test is more appropriate for testing for difference in performance?
3. Is there a statistically significant sex difference in performance? 4. If so, which of the sexes performed better?
GPI %FT PTR GER AIR r TR
GPI 1.0 .64** .45** .35* .40* .06 .08 %FT 1.0 .30* .57** .37* .35* .05 PTR 1.0 .45** .66** -.48** .44** GER 1.0 .80** .49** .46** AIR 1.0 .46** .55** r 1.0 .66**
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Table 4Data set for analysing differences
Performance
School Boys Girls School Boys Girls 1. 75 72 11. 76 55 2. 67 76 12. 77 60 3. 70 70 13. 80 84 4. 86 66 14. 66 60 5. 80 75 15. 77 75 6. 66 60 16. 82 62 7. 88 80 17 62 72. 8. 68 75 18. 74 76 9. 70 75 19. 90 82 10. 66 66 20. 78 78
Application Exercise 5 (Regression Analysis)
1. Think of situations in which you will use a regression technique in analysing the relationship among educational indicators.
2. The relationship between GPI and %FT has been found by researchers to be highly correlated about r = .65 at the .001 level. What is the interpretation of this