1. Module – 10 Research Statistics and Analysis
SCW 952 Academic Writing
Module – 10
Research Statistics and Analysis
Switzerland | Saudi Arabia

2. 6/8/2018
Introduction (2)
Raw data as it is collected is often confusing, hard to make sense of.
Using statistics is a way of making sense of your data, to see if it reveals any patterns or significant relationships between variables.
In order to analyse your data you need to first sort through it, looking at certain characteristics that tell you exactly what kind if data you have.
These two processes of description and analysis are the two different types of statistical processes used:
Descriptive Statistics (DS) is really a way of making sense of your data:
once you have described your data in detail you can then go on to inferential statistics.
In collecting, describing and analysing our data, we are dealing with two types of groups:
our target population which is every individual we are interested in and
our sample which is the group of participants we actually test.

3. 6/8/2018
Introduction (1)
Most of the time we cannot measure all of the population so we use a sample to gain a picture of it. Normally when using DS we are describing the characteristics of our sample data.
When we use Inferential Statistics, we are studying sample data in order to make generalizations about characteristics or relationships present in the target population.
Therefore any sample that we select should be representative of the population, however, any sample will not give a perfectly accurate picture of its corresponding population. .
When using techniques to describe/analyse our data, we talk about statistics when describing samples and parameters describing populations: so the average of a sample would be a statistic, while the average of population would be a parameter.
It is good to keep in mind the difference and relationship between a population and a sample taken from that population, as the fact that samples do not represent populations 100% accurately is taken into account when using certain statistics.

4. 6/8/2018
Using statistics
Carrying out research means the collection of data. Statistics are a way of making use of this data
Descriptive Statistics: used to describe characteristics of our sample
Statistics describe samples
Inferential Statistics: used to generalise from our sample to our population
Parameters describe populations
Any samples used should therefore be representative of the target population
We talked last time about procedures and ways in which observations were made and data was collected. The rest of the course is really about what to do with your data once you’ve got it. Raw data as it is collected is often confusing, hard to make sense of. Using statistics is a way of making sense of your data, to see if it reveals any patterns or significant relationships between variables. In order to analyse your data you need to first sort through it, looking at certain characteristics that tell you exactly what kind if data you have. These two processes of description and analysis are the two different types of stat processes used: DS is really a way of making sense of your data: once you have described your data in detail you can then go on to inferential statistics which.
So (read DS)) In collecting, describing and analysing our data, we are dealing with two types of groups: our target population which is every individual we are interested in and our sample which is the group of participants we actually test. Most of the time we cannot measure all of the population so we use a sample to gain a picture of it. Normally when using DS we are describing the characteristics of our sample data. When we use IS, we are studying sample data in order to make generalizations about characteristics or relationships present in the target population. Therefore any sample that we select should be representative of the population, however, any sample will not give a perfectly accurate picture of its corresponding population. . When using techniques to describe/analyse our data, we talk about statistics when describing samples and parameters describing populations: so the average of a sample would be a statistic, while the average of population would be a parameter. It is good to keep in mind the difference and relationship between a population and a sample taken from that population, as the fact that samples do not represent populations 100% accurately is taken into account when using certain statistics.

5. Descriptive Statistics
6/8/2018
Descriptive Statistics
Statistical procedures used to summarise, organise, and simplify data. This process should be carried out in such a way that reflects overall findings
Raw data is made more manageable
Raw data is presented in a logical form
Patterns can be seen from organised data
Frequency tables
Graphical techniques
Measures of Central Tendency
Measures of Spread (variability)
So Ds as I’ve said are a way of making sense of your data. To begin with you may just have a table of raw scores, that you need to organise in some way. Today we’re going to concentrate on the ways in which you can summarise, organise and simplify your data. DS should complement any use of IS by illustrating any of the significant relationships found: so if you find that manipulating levels var A causes changes in DV B, you want statistics to show this clearly (if changing noise level effects mood: you want to show DS about the two different mood levels).
So we start off with our raw data: what do we want to achieve with DS (read). In order to do this we have a variety of options available including (read types). Apart from the first two types (which are different ways of showing the same info), these should be used in conjunction with each other: as each technique or statistic shows a different important aspect of the data.

6. Plotting Data: describing spread of data
6/8/2018
Plotting Data: describing spread of data
A researcher is investigating short-term memory capacity: how many symbols remembered are recorded for 20 participants:
4, 6, 3, 7, 5, 7, 8, 4, 5,10
10, 6, 8, 9, 3, 5, 6, 4, 11, 6
We can describe our data by using a Frequency Distribution. This can be presented as a table or a graph. Always presents:
The set of categories that made up the original category
The frequency of each score/category
Three important characteristics:
Shape,
Central tendency, and
Variability
Subject #1
Remember
4 things
Subject # 20
Remember
6 things
Ok so let’s consider some actual raw data, and see what type of DS we could produce. (read example). Data is recorded as the number of different symbols a person can remember after being shown a series of 20, with an immediate memory test. There are all the scores in the order in which they were obtained: at the moment telling us very little. This presentation of raw data becomes even more confusing the more data there is. So one of the first things we might want to do is organise this data in some logical way one way is a FD. This presents are data in such a way that categorises scores according to how often they occur: if the data above was shown as a FD, we would be able to see how many people remembered 4 symbols, how many remembered 5 and so on. Again this is especially useful for larger data sets, which can be summarised neatly by a FD.
A FD is commonly represented as a table, which lists the different data scores or measurements, then the number of times that score or measurement is observed in the data set. Presenting data in this way can show us how many people scored what, what the most common score was (an indication of the central tendency of the data set, which I will talk about later), and how spread out the scores are to each other: called the variability of the data set.

7. Frequency Distribution Tables
6/8/2018
Frequency Distribution Tables
Highest Score is placed at top
All observed scores are listed
Gives information about distribution, variability, and centrality
X = score value
f = frequency
fx = total value associated with frequency
f = N
X =fX
So here is the data from the example memory study presented as a FD table: the column labelled X refers to the different scores (so here the numbers of symbols that were remembered) and the f column shows the frequency of these scores (so how many remembered 3 etc.).
A FD table always has the highest score at the top and lists all possible scores down to the lowest score at the bottom. With such a table it is easy to see that most participants remembered around 6 symbols. The last column on the right labelled fx is not necessary for FD, and shows how much of the data set can be attributed to each possible score: so for the score of 11 symbols remembered, 1 person scored 11 so the fx value is 11. For the score of 10 symbols remembered, 2 people remembered 10 which makes fx 2 x 10 =20. The fx column is the (read fx).
You should see from the FD table that using it you can work out certain values: the total number of participants (or N) is gained simply by adding all the values in the f column (which should give up 20). To get the total sum of the scores: which can be obtained by adding all the values of the fx column (which gives us a value of 127)

8. Frequency Table Additions
6/8/2018
Frequency Table Additions
Frequency tables can display more detailed information about distribution
Percentages and proportions
p = fraction of total group associated with each score (relative frequency)
p = f/N
As %: p(100) =100(f/N)
What does this tell about this distribution of scores?
As well as information about scores and frequency of scores: a FD table can also give us more detailed information about the distribution of scores within a data set. Two common measures concern information about the proportion of score frequency. This involves adding columns that shows the information in column, f and fx (i.e. frequency of score and total value associated with frequency) as a proportion or percentage value. So column p shows what proportion of the group remembered each amount of symbols: we can see that a proportion of 0.05 of participants recalled 11 symbols. We convert the frequencies into proportions by simply dividing the frequency by the number of participants so 1/20 = We can also convert the frequencies into % values simply by multiplying the proportion by 100. So we know that 1 participant recalled 11 symbols, this is a proportion of 0.05 (1/20), and 5% (0.05 x 100).
Including the proportions and % is another way of summarising the data: again it shows which score was most common, and how may symbols were recalled how often. So it clearly shows info about the distribution of the scores.

9. 6/8/2018
A frequency distribution FD table can also give us more detailed information about the distribution of scores within a data set.
Two common measures concern information about the proportion of score frequency.
we can see that a proportion of 0.05 of participants recalled 11 symbols. We convert the frequencies into proportions by simply dividing the frequency by the number of participants so 1/20 = 0.05.
We can also convert the frequencies into % values simply by multiplying the proportion by 100. So we know that 1 participant recalled 11 symbols, this is a proportion of 0.05 (1/20), and 5% (0.05 x 100).

10. Grouped Frequency Distribution Tables
6/8/2018
Grouped Frequency Distribution Tables
Sometimes the spread of data is too wide
Grouped tables present scores as class intervals
About 10 intervals
An interval should be a simple round number (2, 5, 10, etc), and same width
Bottom score should be a multiple of the width
Class intervals represent Continuous variable of X:
E.g. 51 is bounded by real limits of
If X is 8 and f is 3, does not mean they all have the same scores: they all fell somewhere between 7.5 and 8.5
FD are a useful tool, but what happens when a set of data covers a much wider range of values than our pervious example? Say you had data with scores ranging from 50 to 100, that’s 50 rows in the X column if you listed each possible score which is not practical. In cases such as these GFD tables are used: we simply group ranges of scores into intervals and record the number of cases that had scores that fall into each particular interval. So this slide shows an example of such a table: the scores range from , so each interval covers 5 points. The GFDt follows the same conventions as the normal FDt: it starts with the highest interval then lists all possible intervals down to the lowest. GFDt should have (read) any more than 10 is too much. Score groups or class intervals as they are called should be based on a simple round number, and all intervals should be the same (so in our example every interval has a range of 5). Bottom score: so if you have an interval range of 5 then the lowest score included in the table should be a multiple of 5.
Something to be aware of is the continuous nature of FD representations. The arrangement of the GFDt means that even though scores are recorded in the same class interval does not mean they are all the same. There are 4 scores recorded in the CI 70-74, one could have 71, one 70, one 73 etc. This is also true of the frequencies in simpler FDt: Class intervals in such GFDt and the X scores on FDt always represent continuous variables: ( a var that is continuously divisible). This means that the data/scores obtained do not represent separate data points but intervals on a continuous scale. This differentiates between apparent limits of an interval and the real limits of an interval. So for example (read 51) when we say 51 we are actually simplifying and using 51 to mean any value that falls within the upper and lower real limits of the value. (read 8)

11. Percentiles and Percentile Ranks
6/8/2018
Percentiles and Percentile Ranks
X values = raw scores, without context
Percentile rank = the percentage of the sample with scores below or at the particular value
This can be represented be a cumulative frequency column
Cumulative percentage obtained by:
c% = cf/N(100)
This gives information about relative position in the data distribution
One final column of very useful info that can be added to a FDt is one that gives context to the frequencies. A FDt gives us a description about a whole set of scores, but it can also be used to give us info about an individual score and its position to other scores (higher/lower) in the data set. We do this by figuring out the percentile rank of a particular score: this is the percentage of people who score at or below that particular score. Say for example we wanted to know what percentage of people remembered 8 or fewer symbols. WE do this by adding a cumulative frequency column, which just means that we add the number of participants for each score as we go along: so 2 people remembered 3 symbols, 3 people remembered 4 symbols: so 5 people remembered 4 or less symbols (3+2). 3 people remembered 5 symbols so we add this to 5 to give us 8 people who remembered 5 or less symbols. To find the percentile ranks we simply convert the values in the cf column into % of the total participant group size by dividing the value by 20 (the number of participants we had) then X 100. So 16 people remembered 8 symbols or less (2 who remembered 8 + the 14 who remembered lower amounts of symbols). If we divide 16 by 20 then x 100 we get 80. So 80% of people remembered 8 or fewer symbols. So say you remembered 8 symbols: your score has the percentile rank of 80%, and is the 80% percentile (called because it is identified by its percentile rank)
So say you remembered 8 symbols:
your score has the percentile rank of 80%, and is the 80% percentile (called because it is identified by its percentile rank)

12. Percentiles and Percentile Ranks - Detail
6/8/2018
Percentiles and Percentile Ranks - Detail
We do this by adding a cumulative frequency column, which just means that we add the number of participants for each score as we go along:
so 2 people remembered 3 symbols, 3 people remembered 4 symbols: so 5 people remembered 4 or less symbols (3+2).
3 people remembered 5 symbols so we add this to 5 to give us 8 people who remembered 5 or less symbols.
To find the percentile ranks we simply convert the values in the cf column into % of the total participant group size by dividing the value by 20 (the number of participants we had) then X 100.
So 16 people remembered 8 symbols or less (2 who remembered 8 + the 14 who remembered lower amounts of symbols).
If we divide 16 by 20 then x 100 we get 80. So 80% of people remembered 8 or fewer symbols.
So say you remembered 8 symbols: your score has the percentile rank of 80%, and is the 80% percentile (called because it is identified by its percentile rank)

13. Frequencies of Populations and Samples
6/8/2018
Frequencies of Populations and Samples
Population
All the individuals of interest to the study
Sample
The particular group of participants you are testing: selected from the population
Although it is possible to have graphs of population distributions, unlike graphs of sample distributions, exact frequencies are not normally possible. However, you can
Display graphs of relative frequencies (categorical data)
Use smooth curves to indicate relative frequencies (interval or ratio data)
So far I have just talked about FD graphs in relation to the example data I presented. Suppose you wanted to construct a graph showing the FD of a population rather than a sample. At the beginning of the lecture I mentioned the differences between samples and populations: and FD graphs are one area where these differences matter. It is perfectly possible to construct any FD graph for population data: a histogram, or polygon as I just talked about, the principles would be exactly the same. The differences lie in the data available for a pop graph: in a sample data set all scores are available, while you may not have access to all the individuals in a population, and so will probably not have scores for the entire population (e.g. a symbol recall data: the target population there may be every adult with a normally functioning memory, not possible to obtain). You may know certain characteristics about a population but it is v rare that you will have all the scores available. You may still draw a graph describing the FD of a population though, you can either:
Use relative frequencies: for categorical (nominal) data you may not know exact numbers, but may know enough to talk about relative frequencies in populations (if you remember for the example data we categorised people into those who could remember p numbers best, those who could remember historical dates best +family dates: we might know that in the general population people are twice as good at remembering family b days than historical dates: you could draw a bar chart representing relative frequency: so it does not matter what frequency value you assign family dates so long as the bar above it is twice as tall as the bar above historical dates).
For populations with interval or ratio data, you could still use a FD graph: if you remember the FD polygon traced the shape of the sample data distribution with a point for each score. For population data you can still use the polygon, but because not all the scores (or points) are known: a smooth curve is drawn to show that these are relative frequencies rather than known scores.

14. Frequency Distribution: the Normal Distribution
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Frequency Distribution: the Normal Distribution
Bell-shaped: specific shape that can be defined as an equation
Symmetrical around the mid point, where the greatest frequency if scores occur
Asymptotes of the perfect curve never quite meet the horizontal axis
Normal distribution is an assumption of parametric testing
So this slide shows an example of a graph for a population of scores: here IQ scores. So we don’t have every single persons IQ level necessary to draw an exact FD polygon. What we do know is that most IQs are clustered around the 100 mark with v few falling at the extreme ends of the scale (i.e. either v high or v low IQ). If you draw these relative frequencies you get a bell shaped curve representing the population distribution. You can see that the vertical axis does not have exact values: what is important on this graph is how the frequency or scores change in relation to each other: so you can clearly see that most people’s IQs fall between , with v few outside this range.
This bell-shaped curve representing the shape of a pop distribution is called a normal distribution: this is a commonly occurring shape for many population distributions. This is a specific shape that can be precisely defined by an equation. ND are characterised by their bell-shaped curve: a curve that is symmetrical (i.e. equal numbers of scores fall either side of the mid point). Many of the data collected are assumed to come from normally distributed pops. This assumption of normality enables us to employ many IS procedures.
So from no on when I talk about distributions I’m referring to the shape the scores make if drawn on such a graph. Rather than bothering to describe a complete FD by either using a table or drawing a graph like the histograms and polygons I showed you earlier, researchers are also able to summarise distributions by listing some main characteristics: for any distribution you want to know: its shape (how do the frequencies of scores change over your different measures), where the centre of the distribution is, and how spread out the scores are.

15. Frequency Distribution: Different Distribution shapes
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Frequency Distribution: Different Distribution shapes
This slide shows some examples of different shapes that D can take. The top left = ND which I’ve already described: majority of scores fall around a mid point with fewer and fewer as the scores get more or less extreme. The top right D shows another type of distribution shape: here the majority of scores fall around two values. The bottom two graphs show D shapes that again cluster around a central value: but unlike the top two graphs they are not symmetrical ( =can draw vertical line through middle and one side is mirror image of each other).
Lets describe these D in terms of what they mean for actual score. Lets imagine that these are all graphs of exam results for different exams, so as the x axis goes along the higher the exam results. The TL graph shows an exam where most people got results around the mid point: v few got v low and v few got v high scores. The TR graph shows 2 peaks, so lots of people got exam results either at one particular lower level (left peak) or a higher level (right peak), with less getting v low, v high or scores in the middle. The BL graph shows that most people got very low scores as the peak of the curve is near the beginning of the x axis, with very few getting high scores (v hard exam). The BR graph is the opposite with the peak of scores near the end of the x axis: so most people scored highly on this exam. D where scores pile at one end or another while the tail of the scores taper off to the other end are called skewed D. So the BL is an example of + skew, so called because the tail of scores tapers off towards the + end of the x axis (looks like p facing upwards), - skew so called because tail tapers off towards – end of x axis. This kind of information is clear from looking at a D.

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17. Measures of Central Tendency
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Measures of Central Tendency
A way of summarising the data using a single value that is in some way representative of the entire data set
It is not always possible to follow the same procedure in producing a central representative value: this changes with the shape of the distribution
Mode
Most frequent value
Does not take into account exact scores
Unaffected by extreme scores
Not useful when there are several values that occur equally often in a set
So by looking at the shape of a distribution we can tell many things about how the scores are placed, where most of the scores lie and where only a few fall. Another useful way of summarising data is to find a single vale that is representative of the whole data set: representative in that it identifies the centre of the data set, and best represents every value in the data set (so could summarise our symbol recall data further by saying that people on average remembered x number of symbols).
There are different procedures for determining the central tendency of a data set, due to the different shapes that D take. If we go back to the previous slide: we are trying to identify the centre of all these D. Only the first one has a clearly defined centre. 2nd: there is a centre, but most of the scores fall elsewhere. For the skewed D: do you go with the central value of the data range or with the most frequent value?
Mode =simplest measure of CT: the most common value: if 8 out of 10 people had IQs of 115 then 115 is the modal value. On a FD curve the mode would be the highest point of the curve. The mode can be used for any scale of measurement (even nominal), you just select the most common score/category. So the mode doesn’t take into account exact scores, just score frequency. Because of this, it is unaffected by data sets that contain scores that are at one or the other extreme end of the measurement set: for example if 8 out of 10 people had IQs of 115, then the mode would still be 115 the IQ range for those ten people was or Can be more than one mode (read)

18. Measures of Central Tendency
6/8/2018
Measures of Central Tendency
Median
The values that falls exactly in the midpoint of a ranked distribution
Does not take into account exact scores
Unaffected by extreme scores
In a small set it can be unrepresentative
Mean (Arithmetic average)
Sample mean: M = X Population mean:  = X
n N
Takes into account all values
Easily distorted by extreme values
The second measure of CT is the Median: this is the score that divides the data set exactly in half. #so if you remember back to percentile ranks then the median would be equivalent to the 50% percentile (meaning half of the participants have score at or below the median value). So if we had a v small data set of scores 1,1,and 4, then 1 would be the median value (as it is the middle score of the sample: there are equal numbers of scores lower and higher than it. For a sample with an odd total number of scores we simply select the middle value as the median: for even numbers of scores we select the middle two values (such that there are equal numbers of scores lower and higher than the two scores), add them together and divide by two. Like the mode the median does no take into account the exact value of scores, just their relative position to each other: therefore again it is unaffected by extreme scores in a data set.
The last measure of CT is the mean: also called (read) This is computed by adding all the scores together and dividing this total by the number of scores. This is what this equation shows. Different notations are used to depict sample means and pop means: for a sample =M or x bar, for a pop by the Greek letter “myoo,” but the equations are exactly the same. The mean can be thought of as the balance point of a D: if we go back to our v small sample of 3 scores: 1,1, and 4, the mean is 6/3 is 2. We can see that the total amount of scores above the mean is exactly the same as the total amount below the mean: the mean is 2, both scores of 1 making an amount of 2 below the mean value, the last score is 4 which is 2 above the mean value. So the mean takes into account the actual values of the scores: if the scores are extreme the mean will reflect this unlike the mode and the median: say if we went back to the sample 1,1, and 4, if we changed the last score to 5 instead of 4 only the mean value would change.

19. Measures of Central Tendency
6/8/2018
Measures of Central Tendency
For our set of memory scores:
4, 6, 3, 7, 5, 7, 8, 4, 5,10
10, 6, 8, 9, 3, 5, 6, 4, 11, 6
Mode = 6: Median = 6: Mean = 6.35
The mean is the preferred measure of central tendency, except when
There are extreme scores or skewed distributions
Non interval data
Discrete variables
So if we go back to our original example data: our recall scores for symbols: we can figure out all three measure of CT. The mode would be the most frequent score: 4 people scored 6 so 6 is the mode. The median score would be the score at the exact middle of the D: there are 20 scores so we order the scores from lowest to highest. We then take the middle two scores the 10th and 11th scores (so there are nine scores above and below) which are both scores of 6: we add these together and divide by two which gives up a median of 6. Finally we add up all the scores and divide this by 20 (sample size) to find out our mean which is 6.35.
For most DS the mean (read). Because the mean is affected by the actual values of scores there are certain times when it becomes too distorted by extreme values to be considered representative of the data set as a whole (say for example if you had a set of 10 scores, 9 of which were 1 + the last = 50: the mean for this sample would be 5.9 which is not representative of the set of scores which the vast majority of are much lower). This is also an example of skewed data where all the scores are piled at one end of a measurement scale apart from a few the distort the mean. Because the mean is arithmetical it cannot be used on non numerical data

20. Central Tendencies and Distribution Shape
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Central Tendencies and Distribution Shape
It is possible to represent the CT (whatever type) on the FD polygon by bisecting the curve with a straight line to represent your mean median or mode. With a ND (symmetrical, bell-shaped curve) the mean, median , and mode will all have the same value and will be represented by the exact centre of the D. For symmetrical D the mean and median will always be exactly in the centre, and will always be the same value.
On skewed D shown here the position of the 3 measures differs slightly. The mode will always be at the highest point of the curve (representing the most frequent score); the median will be exactly in the middle of the D (the middle position). On + skew (the L graph) where the majority of scores are at the lower end of the scale, this means that the mode will be the lowest value, followed by the median. Then mean score will be affected by the few extreme higher scores and so will be the highest values of the 3 measures of CT. On the –skew where most of the scores are at the higher end of the scale, the opposite pattern is observed.

21. Describing Variability
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Describing Variability
Describes in an exact quantitative measure, how spread out/clustered together the scores are
Variability is usually defined in terms of distance
How far apart scores are from each other
How far apart scores are from the mean
How representative a score is of the data set as a whole
So far we have considered ways of describing the shape of a D and its central point. There is one other important characteristic of a data distribution set: how spread out the data scores are: whether they are clustered closely together or not. This would be a measure of variability for our data (read Last point : if an individual score is close to the mean or not )
The diagram shows how V can be important: these two distributions have the same shape: they are both symmetrical, and ND. They have the same mean value. They don’t however have the same V. Say these are graphs showing IQ from two different samples of people. In the L graph the spread of the scores is much smaller than the R hand graph: so for the L hand graph the IQs are clustered together over a smaller spread from say , while the R hand graph the IQs are spread out over a much wider range: say

22. Describing Variability (V)
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Describing Variability (V)
How spread out the data scores are: whether they are clustered closely together or not.
This would be a measure of variability for our data (read Last point : if an individual score is close to the mean or not )
The diagram shows how Variability can be important: these two distributions have the same shape: they are both symmetrical, and ND.
They have the same mean value.
They don’t however have the same Variability.
Say these are graphs showing IQ from two different samples of people.
In the Left graph the spread of the scores is much smaller than the Right hand graph: so for the Left hand graph the IQs are clustered together over a smaller spread from say , while the Right hand graph the IQs are spread out over a much wider range: say

23. Describing Variability: the Range
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Describing Variability: the Range
Simplest and most obvious way of describing variability
Range = Highest - Lowest
The range only takes into account the two extreme scores and ignores any values in between. To counter this there the distribution is divided into quarters (quartiles). Q1 = 25%, Q2 =50%, Q3 =75%
The Interquartile range: the distance of the middle two quartiles (Q3 – Q1)
The Semi-Interquartile range: is one half of the Interquartile range
The simplest way to get a measure of V is to quantify the range of scores: within what interval do all scores fall. This is called the range, and is found simply by subtracting the lowest score from the highest score. So back to our symbol recall data: the lowest score was 3 symbols, while the highest score was 11 symbols therefore the range for this data set is 9 symbols (all scores were within 9 symbols of each other). The range is 9 and not 8 because when computing the range you must take into account the real limits of the scores: our range for the memory data is not obtained from subtracting 8 from 11 but subtracting 2.5 (the lower real limit of 3) from 11.5 (the upper real limit of 11).
While the range is the most obvious measure of V, it is completely determined by the two extreme scores of a data set, and takes no notice of any scores in between, which is not representative of the spread of the scores as a whole (so if 10 scores 9 of 1 and last one =50, the range would be 50 despite the scores all being clustered at 1). One way around this problem is by dividing the distribution by 4 or into quartiles. QI is the score with exactly 25% of all scores below it, Q2 = 50% of all the scores ( and so is the median), and Q3 = 75% of all scores. The interquartile range is the distance between the Q3 and Q1 so the middle half of the distribution, which aims to miss out any extreme scores that may cause an unrepresentative range. Another common measure is the SIR which is the IR divided by 2 (and so measures the range of the middle 25% of the distribution). Both these modified ranges ignore either ends of the distribution, and so aim to be more representative of how spread out the scores are by concentrating on scores in a more central position. However, both the IQR and the SIQR ignore some of the distribution, and none of the range measures take into account actual distances between scores.

24. Describing Variability: Deviation
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Describing Variability: Deviation
A more sophisticated measure of variability is one that shows how scores cluster around the mean
Deviation is the distance of a score from the mean
X - , e.g = 3.65, 3 – 6.35 = -3.35
A measure representative of the variability of all the scores would be the mean of the deviation scores
(X - ) Add all the deviations and divide by n n
However the deviation scores add up to zero (as mean serves as balance point for scores)
A much more sophisticated measure of V is one that that takes every score into account. V is meant to tell us how spread out scores are, one way of doing this is by taking a measure of how far each score deviates from the central point of the distribution or the mean. When I say dev I mean this: how far a score dev from the mean value. To obtain these deviations, we take the mean away from each score: so from our symbol recall data lets start with our highest score of 11 symbols remembered. The mean no of symbols remembered was 6.35, so we simply subtract 6.35 from 11 to give us a deviation of So the score of 11 is 3.65 symbols above the mean value. To calculate the deviation of the lowest score (3 symbols), we do exactly the same thing: subtract from 3 which gives us a deviation of -3.35: the lowest score was 3.35 symbols below the mean (the – sign tells us that the score is below the mean while + = above the mean).
So we can calculate deviations for each score using the same procedure of subtracting the mean from the score. To get a single measure of V for the whole D, we could calculate the average deviation: this would show us how far away on average each score is from the mean. So how would we get this average? We could add all the deviations together and divide this total by the number of participants: this would be a straightforward replication of the method used to calculate the mean. However , we can’t just do this because of the nature of deviations: each deviation tells us the distance of that score from the mean: but the mean is the balancing point of the D and so has equal distances below and above the mean. Therefore the dev scores will add up to zero and we can’t divide 0 by anything, so we cannot get an average this way. The devs totalling 0, are caused by the different positive and negative values (because there are equal distances below and above the mean) cancelling each other out. So we need to remove the signs in order to be able to calculate the average dev.

25. Describing Variability: Deviation
6/8/2018
Describing Variability: Deviation
When we say dev, we mean this:
How far a score dev from the mean value?
To obtain these deviations, we take the mean away from each score: so from our symbol recall data lets start with our highest score of 11 symbols remembered. The mean no of symbols remembered was 6.35, so we simply subtract 6.35 from 11 to give us a deviation of So the score of 11 is 3.65 symbols above the mean value. To calculate the deviation of the lowest score (3 symbols), we do exactly the same thing: subtract from 3 which gives us a deviation of -3.35: the lowest score was 3.35 symbols below the mean (the – sign tells us that the score is below the mean while + = above the mean).

26. Describing Variability: Deviation (Cont..)
6/8/2018
Describing Variability: Deviation (Cont..)
We can calculate deviations for each score using the same procedure of subtracting the mean from the score. To get a single measure of V for the whole D, we could calculate the average deviation: this would show us how far away on average each score is from the mean. So how would we get this average? We could add all the deviations together and divide this total by the number of participants: this would be a straightforward replication of the method used to calculate the mean. However , we can’t just do this because of the nature of deviations: each deviation tells us the distance of that score from the mean: but the mean is the balancing point of the D and so has equal distances below and above the mean. Therefore the dev scores will add up to zero and we can’t divide 0 by anything, so we cannot get an average this way. The devs totalling 0, are caused by the different positive and negative values (because there are equal distances below and above the mean) cancelling each other out. So we need to remove the signs in order to be able to calculate the average dev.

27. Describing Variability: Variance
6/8/2018
Describing Variability: Variance
To remove the +/- signs we simply square each deviation before finding the average. This is called the Variance:
(X - )² = = 5.33 n
The numerator is referred to as the Sum of Squares (SS): as it refers to the sum of the squared deviations around the mean value
So here is a table showing the scores from the symbol recall data and each scores deviation (in the middle column): difference from You can see that all scores below the mean have –devs while all the scores above the mean have +devs. These values then cancel each other out leading to a sum of 0. In order to overcome this we simple square each dev, which will remove the signs: all the devs become +. So the R column shows each dev squared. We can then use the total of these squared devs to find the average dev. So add up all numbers in R column, then divide this total by the no of participants (20) = 5.33 = This value is called the Variance
The Variance is the mean squared deviation, and worked out exactly as we would for the mean score: to find the mean we add up all the individual scores and divide this by the no of the sample. In the case of the Var we add up all the individual squared devs: the top part of the equation describes this. We call this the sum of the squared devs or sum of squares (written as SS) (read) So we can say that in order to find the Var we divide the SS by the number of Pts.

28. Describing Variability: Population Variance
6/8/2018
Describing Variability: Population Variance
Population variance is designated by ²
² = (X - )² = SS
N N
Sample Variance is designated by s²
Samples are less variable than populations: they therefore give biased estimates of population variability
Degrees of Freedom (df): the number of independent (free to vary) scores. In a sample, the sample mean must be known before the variance can be calculated, therefore the final score is dependent on earlier scores: df = n -1
s² = (X - M)² = SS = = 5.61
n n
So we can say that in order to find the Var we divide the SS by the number of Pts. This formula is slightly different for working out the Var of pop or samples. For a population Var (shown by the lowercase sigma squared) we just follow the procedure I just explained: find the sum of every squared dev then divide this by the number of participants. Working out the Var for a sample is slightly different: often sample var is used as an estimate for pop var, when such info about a pop is unavailable. However for pop var you calculate dev by subtracting each score from the pop mean: but for sample var you will subtract each score from the sample mean. As sample means are usually not identical to pop mean, this will means that calculating pop var and sample var (when sample is taken from that pop) will result in different end Var. In order to be able to use sample Var as an estimate of pop var a simple change has to be made to the var equation. Sample var tends to underestimate pop var: without going into great detail this partly comes from the fact that a sample has a set number of scores, and the sample mean must be known before calculating devs. This means that in any sample all the scores are free to vary except the last one, which is dependent on the other score values. Think of a sample of 3 scores with a mean of 10. the first two scores can be any number, say 1 and 2, but the last score cannot just be assigned because it must be a value that makes the overall mean 10 (so it has to be 27). This sample only has 2 independent score: 2 out of 3 or n-1. The number of independent scores in a sample are called Df.
So in order to calculate sample var that is not an underestimate (or biased) estimate of pop var, instead of dividing the SS by the total number of scores, we divide the Ss by the number of scores in the sample that are free to vary (or n-1). So for our symbol recall data, we already calculated the var as 5.33, but that was using the formula for pop var. We are calculating the sample var, so we divide the sum of the squared dev by n-1. Sample var is shown by a lowercase s squared to distinguish it from pop var.

29. Describing Variability: the Standard Deviation
6/8/2018
Describing Variability: the Standard Deviation
Variance is a measure based on squared distances
In order to get around this, we can take the square root of the variance, which gives us the standard deviation
Population () and Sample (s) standard deviation
 = (X - )² N
s = (X - M)²
n - 1
So for our memory score example we simple take the square root of the variance:
= 5.61 = 2.37
So now we know how to calculate variance. However one problem with var is that it is based on squared distances from the mean which can be hard to think of. So for our symbol recall data we have a mean value of 6.35 symbols remembered, with a var of 5.61 (which means that scores were an average of 5.61 squared symbols away from the mean). This is a bit hard to quantify: so we could take the square root of the var so that the distances are not based on squared units. The square root of the var is called the SD = the typical or standard distance of the scores from the mean.
(read notation) As with equations for sample var we divide the SS by the df n-1 to find the sample SD.
So inorder to find the SD for our symbol recall data what do we do?
Subtract the mean value from each score: these are our devs
Square each of the devs then add these together
Divide this total by n-1 (20-1): this gives you the var for our data. Square root this value to find the SD. So var was 5.61: SD is SRof 5.61 = We can now say that in the symbol recall data, the scores were typically 2.37 symbols away from the mean of 6.35 (gives you an ides of spread of scores).

30. Describing Variability
6/8/2018
Describing Variability
The standard deviation is the most common measure of variability, but the others can be used.
A good measure of variability must:
Must be stable and reliable: not be greatly affected by little details in the data
Extreme scores
Multiple sampling from the same population
Open-ended distributions
Both the variance and SD are related to other statistical techniques
The SD (and var) is the most commonly used measure of V. In some sits the IQ range and SQ range can be used: if there are a few v extreme values the AD will be affected by this. The IQ range ignore the top and bottom ands of the D and so will not be affected.
If you take several different samples from the same population the SD and var will tend to give similar measure of V for each sample. They ar said to be stable under sampling.
Some D are open-ended: does not have specific boundaries for the lowest or highest scores: for example you may have undetermined scores: e,g, a prob solving study where one pt has taken an extremely long, undetermined time to solve the problem. In such a case only the SIQ range is available as a measure of V.
One reason that SD and Var are so commonly used is that (read). This is because SD, var and stat techniques are based on squared differences.

31. Descriptive statistics
6/8/2018
Descriptive statistics
A researcher is investigating short-term memory capacity: how many symbols remembered are recorded for 20 participants:
4, 6, 3, 7, 5, 7, 8, 4, 5,10
10, 6, 8, 9, 3, 5, 6, 4, 11, 6
What statistics can we display about this data, and what do they mean?
Frequency table: show how often different scores occur
Frequency graph: information about the shape of the distribution
Measures of central tendency and variability
So I’ve talked about three main characteristics of describing data. Let go back to example data and try to piece them all together. So if you remember (read). We had our raw data –what can we display about the data using the techniques I’ve described?
We can describe the shape of the distribution of the scores: this can be achieved by the FD table or graph
We can describe where the mid point of the D is (using mean median and mode)
We can describe the spread of the scores: how clustered they are to each other, and the typical distance the scores are from the midpoint: using SD

32. Descriptive statistics
6/8/2018
Descriptive statistics
So here is our organised data: the FD table shows the frequency of each score, and the proportion and percentage of people who remembered each observed score. The FD polygon shows us the shape of our D: we can see that most people had scores around the middle of the score range. The red circle is the position of the median and the mode

33. Homework Read critically the following thesis’ sample part:
Are you able to understand the statistics?
Are you able to distinguish any problems or issues on presenting the data?
What would you do better?