The aim of this investigation is to gain statistical information to show authorship of a text.
AS Mathematics: (AQA) Statistics Coursework
DESIGN
Introduction:
The aim of this investigation is to gain statistical information to show authorship of a text. For this investigation, I will use two pieces of text in order to investigate authorship. In order for the investigation to be valid, the two pieces of text I need to use should have a different theme attached to them. By theme, I mean they need to be different in a broad way i.e. different genre, different age readers.
I had a number of different texts to compare but I decided to use one adult text and one child text as this will give me a more obvious variation and expectation.
For this investigation I will be calculating the mean of the distribution for both populations. I will then be able to calculate the standard deviation and variance, and I will be using the unbiased estimator for both populations. I will calculate the standard error and confidence intervals for both populations.
My data will be represented using frequency distribution tables and these can show the trends of a frequency distribution graph. The normal distribution diagrams will also be used for the confidence intervals representation.
Population:
In a statistical enquiry, you often need information about a particular group. This group is known as the POPULATION and it could be small, large or infinite. The population for my investigation is the all the words of each separate book.
Sampling:
Sampling is the selection of individual members of a population. The advantage of taking a sample is that it is cheaper, quicker and the results are easier to analyse and the appropriate for this type of investigation. Unfortunately, it does have some disadvantages that are difficult to avoid as the results may include natural variation or bias, and so may not be representative of the whole population and thus the results are meaningless.
The following are important factors to take in consideration when choosing a sample for this type of investigation:
==> The sample size must be large enough so that the results are more accurate. A very small sample may not represent the rest of the population. So I must make sure that any sample I take is large enough to be representative of the population as a whole. So in order to get more accurate results and for the data I collect to be representative of the whole population, I am going to take 50 samples in total for both the books.
==> The sample should be taken at random. If a random sample is not taken, then my results may be biased. If I choose which page and which line, I wanted to count the number of words then I will end up with data, which is unrepresentative. So in order to get a set of data, which is representative; I used the RAND function on my Casio calculator to get the random page number, line number and word number.
Sample size and method:
For this investigation, I have decided to arbitrarily select 50 pages from each text. The main reason the pages are random is to avoid any biased results that would give inaccurate results and therefore give a bad interpretation of the population parameters.
Fifty pages is also enough pages to get an overview of the means, standard deviation etc. A larger sample is creates more accurate results than a smaller sample and 50 pages is sufficient enough to determine accurate representations of specific parameters and it is also a large enough sample for a theory I am going to use later in the coursework.
Method:
For this investigation, I am finding out whether it possible to gain information about authorship of a text. I will be using and adult text and a child text. The adult text that I will be using is 'Watchers' by Dean R Koontz which is a book that consists of 507 pages. The child text I will be using is 'Million Dollar Egg' by Roderick Hunt. This book consists of 45 minus three pages at the beginning of the text. I will select 50 random pages from each book. I will then select a random line and word on each of these pages.
How I will select the pages:
I will select 50 random pages from each book by using the RAND function on my calculator. Once I have 50 random pages for each book, I will select a random line for each random page. From the random line, I will then finally select a random word from each line.
All calculators have a random key, which can be used to generate random numbers. On the majority of scientific calculators they are decimals give correct to 3 decimal places. By multiplying by a fixed value, you can generate values up to a fixed maximum.
e.g. If there are 50 pages in a book, and a random page is required, if RANDOM 50 is typed into a calculator, it will generate numbers that can be rounded 0 to 49, so to choose an accurate random page of 50 pages, I would have to type (RANDOM * 50) +1.
After typing (RANDOM*50) + 1, I will need to consider the numbers before the decimal point, as I will need to round of the number to give an integer as, book pages are consecutive whole numbers, not decimals:
e.g. 304.9 will be 305, 76.3 will be 76 (to 1 d.p)
Using the RAND Function:
(X, Y and Z would be constants corresponding to the page, line and word number total)
e.g. X × RAND function (X is number of pages in each book)
Y × RAND function (Y is number of lines on X)
Z × RAND function (Z is the number of words on Y)
The number of pages for each book will obviously be the same. However the number of lines varies as each chapter ends, and the number of words on each line varies, as larger words use up more space causing fewer words per line and vice versa. For this reason I will obviously count the number of lines on each page and times this with the RAND function to make the random number accurate each time. This process will also be used for the word selection procedure.
Encountering difficulties
When selecting the random words, by chance I could encounter a word with hyphens or any other grammarcal expressions. To avoid any an unfair investigation I have chosen to ignore these types of words or expressions and instead another random word should be selected. i.e. Mr. is not considered as a word. I will also ignore conjunctions. Before I collect my data I must also consider one other thing to do with the two texts that could affect the reliability of my results. The main one was to do with punctuation. I would only be ...
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Encountering difficulties
When selecting the random words, by chance I could encounter a word with hyphens or any other grammarcal expressions. To avoid any an unfair investigation I have chosen to ignore these types of words or expressions and instead another random word should be selected. i.e. Mr. is not considered as a word. I will also ignore conjunctions. Before I collect my data I must also consider one other thing to do with the two texts that could affect the reliability of my results. The main one was to do with punctuation. I would only be counting letters. This meant things like full stops, apostrophes and even numbers left in numerical form where excluded. Abbreviations such as 'Dr' were counted as they were seen; i.e. Dr equals 2 letters and not 6. I also started on the first full word of each randomly generated line.
Another problem with the adult text I am using is that before a chapter it has blank pages which are counted as pages so if by random I come upon of those pages I will generate another random number to give me a valid page for my investigation and minimise inaccuracies.
Hypothesis
The two pieces of text I will be using are:
==> 'Million Dollar Egg' by Roderick Hunt (CHILD TEXT)
==> 'Watchers' by Dean R Koontz (ADULT HORROR THRILLER)
I predict that there will be on average more letters per word in the adult horror text, and fewer in the child text. The reason for this prediction is that I expect the author of the adult text to use longer, descriptive words as the reader is expected to have a large vocabulary, where as on the contrary, the child text would have smaller words, as the expected reader I a child with a small vocabulary.
Therefore, the mean in 'Watchers' should be loner, with a larger standard deviation and variation between the words because of the use of a large varying vocabulary. After applying the central limit theorem I will calculate the standard errors and confidence intervals, and I predict the child text to have a smaller confidence interval than the large as the population of the child text is small and the sample should be relatively more accurate at representing relevant parameters more than the adult text.
Assumptions
The distribution of the parent population is normal. We have to assume that the distribution of the sample is also normal to have accurate results. I have assumed that a sample size of 50 would give me a normal distribution. This can also be further explained using the central limit theorem which I will discuss in detail later.
Statistical application
==> After collecting the data, I will set the results out in a tally chart because it is easier to understand and analyse.
==> I will use the central limit theorem because it will make it easier to make predictions about the distribution of the sample mean even if the distribution of the parent population is not known.
==> I will draw a frequency distribution graph to show the distribution of the data for both books. I will also work out the mean to work out the average. The variance and standard deviation would help me measure the spread of the data. If I work out the standard error then it will help me to be confident in my estimate of the population mean.
==> In addition, I am going to use unbiased estimation because this will help me to find the variance of the parent population.
Statistical Theories
I will be using a number of Statistical Theories. The first is the 'Central Limit Theorem'. This means that if our population is not normal it can still become approximately normal providing the sample size is large enough. The mean of our sample means will be approximately equal to the mean of the parent population, which in this case is the book. The variance of the distribution of our sample means will be approximately the variance of the parent population divided by our sample size. It is possible to state the 'Central Limit Theorem' symbolically. This is shown below:
if X ~ (unknown)(?,?²) then ?n ~ N (? , ?²)
A good sample size to use is n ? 30. This means that our sample size of 50 words per population is enough.
I will also be using an estimate of the population mean (x). This will become the sample mean, which is an unbiased estimator. I will also do an estimate of the population variance. The sample variance is biased by itself so we will use the formula:
Sample variance x ( )
I will be using the normal distribution. This will be done using the distribution of sample means. By using this distribution I will be able to find out confidence intervals. I will be using the confidence intervals for 68%, 90%, 95% and 99% depending on which ones are suitable for my results. However I will be using 95% definitely. I will get my confidence intervals from:
X - 1.96 (s.e) < ? < X + 1.96 (s.e)
This would be used to find out the 95% confidence interval. The 1.96 comes from the normal distribution tables. In the tables I looked for 0.9750. This is 97.5%. This may seem strange but the reason I found this and not 95% was to do with the symmetry of the normal curve. If I find 97.5% it leaves 2.5% on either end of the curve. This adds up to 5% so 95% is left over. This is sown more clearly in the diagram on the diagram below.
The standard error (s.e) is the standard deviation of the distribution of the sample means. It is found using the formula:
s.e. =
Data Representation:
Here are all the relevant blank tables I will be using for both the adult text and the child text.
st Table for recording of relevant data:
Word
Page
Word
Word Length
2
49
50
2nd Table to show frequency distribution of data:
No. of letters(x)
Tally
Frequency (f)
2
3
4
5
6
7
8
9
0
1
2
Graph to show the distribution of the data
i.e.
Data Collection and Statistical Analysis:
Raw Data for the adult text:
Word
Page
Word
Word Length
409
happening
9
2
280
eight
5
3
365
einstein
8
4
281
of
2
5
383
in
2
6
92
instinctual
1
7
415
within
6
8
425
have
4
9
42
window
6
0
238
there
5
1
401
tracer
6
2
05
wrench
6
3
391
edge
4
4
381
her
3
5
58
was
3
6
97
frightened
0
7
207
four
4
8
99
meaning
7
9
3
descended
9
20
296
but
3
21
225
you
3
22
02
food
4
23
335
in
2
24
03
once
4
25
346
button
6
26
351
three
5
27
486
making
6
28
25
first
5
29
82
you
3
30
361
observing
9
31
95
not
3
32
441
same
4
33
5
animal
5
34
345
get
3
45
333
behind
6
46
58
swing
5
47
318
there
3
48
222
could
5
49
28
the
3
40
49
carefully
9
41
389
woman
5
42
343
bottoms
7
43
281
peculiar
8
44
379
dilworth
8
45
10
vulnerable
0
46
501
accepted
8
47
370
redwood
7
48
405
paintings
9
49
87
physician
9
50
253
intelligence
2
Frequency Distribution Table and graph for adult's text:
No. of letters(x)
Tally
Frequency (f)
0
2
I I I
2
3
I I I I I I I I I
9
4
I I I I I I
6
5
I I I I I I I I I
9
6
I I I I I I I
7
7
I I I
3
8
I I I I
4
9
I I I I I I I
7
0
I I
2
1
I
2
I
The distribution is not normal and I will discuss how a certain theorem acknowledges this.
Raw Data for Children's Text:
Word
Page
Word
Word Length
24
mum
3
2
8
to
2
3
7
was
3
4
29
their
5
5
31
trouble
7
6
3
she
3
7
8
was
3
8
45
there
5
9
5
the
3
0
9
very
4
1
37
a
2
38
he
2
3
20
it
2
4
45
to
2
5
26
and
3
6
5
eggs
4
7
25
chris
5
8
30
friends
7
9
35
archie
6
20
33
to
2
21
40
yellow
6
22
2
hands
5
23
0
out
3
24
43
house
5
25
6
on
2
26
42
jacket
6
27
4
was
3
28
38
oh
29
7
said
4
30
25
there
5
31
35
for
3
32
37
chris
5
33
36
cat
3
34
0
coops
5
45
43
half
4
46
40
of
2
47
6
the
3
48
5
place
5
49
33
bring
5
40
9
six
3
41
45
picture
7
42
46
lots
4
43
27
sing
4
44
41
down
4
45
4
glass
5
46
36
the
3
47
23
too
3
48
37
it
2
49
4
the
3
50
26
want
4
Frequency Distribution table and graph for Children's Text:
No. of letters(x)
Tally
Frequency (f)
I I
2
2
I I I I I I I I
8
3
I I I I I I I I I I I I I I I
5
4
I I I I I I I I
8
5
I I I I I I I I I I I
1
6
I I I
3
7
I I I
3
8
9
0
The distribution is not normal and I will discuss how a certain theorem acknowledges this
Mean
The mean is unbiased and is the weighted average of the number of letters per word and can be presented as:
x = Sfx
Sf
x = mean
Sfx = the total of frequency times the interval
Sf = the total of frequencies
Adult Text
Child Text
Variance
To calculate the variance the following formula is used:
s² = Sfx² - x²
Sf
Adult Text
Child Text
Standard Deviation
s = Sfx² - x²
Sf
s = v (variance)
Adult Text
Child Text
Unbiased Estimation
A sample statistic is called an unbiased estimator if its expected value is equal to the population it is estimating. Otherwise the statistic is known as biased.
The estimate is biased if the average (or expectation) of a large number of values taken in the same way is the true value of the parameter. There may be several ways of obtaining an unbiased estimate but the best (most efficient) estimate is the one with the smallest variance.
The Mean
The sample mean, x, is an unbiased estimator of the population mean. The mean of all possible sample means, with the sample size n, is equal to the population mean, so the calculated means are already unbiased.
The Variance
The sample variance is a based estimator of the population variance. S2 is an unbiased estimator of the population variance. They are given n by the following formula:
Adult Text
Child Text
Standard Error
s.e. = s
vn
Where s = standard deviation.
vn = Square root of the number of samples.
Adult Text
Child Text
Confidence Intervals
As I briefly stated before, I am going to find out the confidence intervals for the two populations.
The confidence interval is an interval that has a specified probability of including a parameter. The interval is usually written (a, b) and the end-values, a and b, are known as confidence limits. The probabilities most often used in confidence intervals are 90% and 95% percent.
The interval that I will construct uses the values of the mean o a random sample of the size n taken from the population. The mean is denoted by x.
Before constructing the interval for means, it is essential to ask the following questions:
==> Is the distribution of the population normal or not?
==> Do you know the variance of the population?
==> Is the sample size large or small?
Confidence interval (mean), of the population
When calculating confidence intervals, it is often the case that the population variance is not known. Provided that the sample size n, is large (n > 30) it is permissible to use the unbiased variance, the best unbiased estimate for the variance.
Ideally the distribution of X (e.g.) should be normal, but an appropriate confidence interval can also be given when the distribution of X is not normal, providing n is large (n>30 according to the central limit theorem)
Formula:
Providing that n is large (n>30), a 95% confidence interval for the mean is
Where:
Critical z-values:
The 1.96 from the other page is the z-value for a 95% confidence interval, but where does it come from?
The z-value in the confidence interval is known as the critical value and is obtained for different levels of confidence as follows:
==> In a 90% confidence interval, the upper tail probability is 0.05, so the lower tail probability is 0.95
==> In a 95% confidence interval, the upper tail probability is 0.025, so the lower tail probability is 0.975
Summary
Table of critical value
The z-values can be derived from the table below depending on the probability percentage required. In some tables the most commonly used critical z-values are summarised as follows:
(If Z ~ N (0,1) then, for each value of p, the tables gives the value of z such that P ( Z < z ) = p )
p
0.75
0.9
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
z
0.647
.282
.645
.96
2.326
2.576
2.807
3.09
3.291
==> For a 90% confidence interval, (P Z < z ) = 0.95, p=0.95 gives z = 1.645
==> For a 95% confidence interval, (P Z < z ) = 0.975, p=0.975 gives z = 1.96
Confidence Intervals (90% and 95%) for Adult Text
Confidence Interval (90% and 95%) for child Text
Central Limit Theorem
If the sample size is large enough, then the sample means is approximately normal, irrespective of the distribution of the parent population. The mean of the distribution of the sample means is approximately equal to the mean of the parent population. The variance of the distribution of the sample means is approximately the variance of the parent population divided by the sample size. These approximations get closer as the sample size gets bigger. These important results are known as the central limit theorem.
If X ~ (unknown) ( µ , s² ) then Xn ~ N ( µ , s²/n)
Therefore for samples taken from a non-normal population with mean µ, and variance s², by the central limit theorem, the distribution is approximately normal and:
Provided that the sample size, n, is large (n > 30)
Adult Text
Child Text
Conclusion:
After completing all my calculations, I managed to get the results that I wanted. The confidence intervals did not overlap allowing me to confidently state that that mean of the parent population was greater in the adult text, than in the child text, which coincides with my prediction, and I can confidently say my hypothesis was right.
The frequency distribution for both sets of data was not normal, but appeared to be more skewed to one side. This could have been due to small sample size used, which was unrepresented of the real population distribution. However, the central limit theorem, states that if the sample size is large enough, then the sample mean is approximately normal, irrespective of the parent population.
The variance and standard deviation, were higher in the adult book, just as expected and stated in the hypothesis, as a possible reason for this could have been due to the fact, the adult book (Watchers), uses a variety of vocabulary, where as the child text does not
The standard error for the child texts was lower than the standard error for the adult text. This showed that the adult text had more variation in the sample means values than the child's text.
Using the unbiased estimator I worked out the variance and the standard deviation of the parent population by just using the sample data. When I worked them out I found that the variance of the sample and the parent population were very alike. The standard deviations of the sample data and the parent population followed a similar pattern.
Interpretation and validation:
Mean
The mean is the average of the sample of data collected. The mean of the sample of data collected for Adult text is 5.66. The mean of the sample of data collected for child text is 3.78. This shows that the adult text uses larger words according to my sample.
Variance and Standard Deviation
The variance is a measure of spread. The variance for adult text is 5.48. The standard deviation for the adult text is 2.3409. The variance for the child text is 2.25 . The standard deviation is 1.5. This shows that the adult text had a wider spread of words used and it seems that the difference in spread is nearly two, implying that the vocabulary on the adult text was twice as varied as the vocabulary used in the child text, which coincides with my hypothesis that the adult text uses a more broader selection of words ranging from long standard words to short and potent words, where as the child text used a simple range of words that a 10 tear old is expected to be familiar with.
Standard Error
The standard deviation of the distribution of the sample mean is called the standard error of the mean. The larger the sample size is, the smaller the standard error. The standard error of the mean for the adult text is 0.33105 and the standard error of the mean for the child text is 0.2121. The standard error is possibly more smaller due to the fact that the child text only contains a small amount of words compared to the adult text and thus a small sample will reflect the parent population more precisely on a smaller parent population and in this case, the child text
Confidence Intervals
==> For the adult text, I am 95% confident that the parent population mean is between 5.01142 and 6.3088. This means the mean value of the parent population lies in between these two values.
==> I am 90% confident that the parent population mean is between 5.1154 and 6.2045.
==> For the child text, I am 95% confident that the parent population mean is between 3.36428 and 4.1957.
==> I am 90% confident that the parent population is between 3.4310 and 4.1289.
The confidence intervals did not overlap and statistically proved my hypothesis that the adult text had a higher mean than the child text. Although the hypothesis was proven right, I expected the difference in the mean between the both, to be much higher than expected. The adult seemed to have an average word length of 6 letters, where as the child text was 4 letters. These values are not far away from each other as I expected the adult text to have a higher mean, i.e. 8-9 letters as I thought the author would use long evocative words to generate the horror atmosphere. This was not confirmed by my sample so this can be extended as a larger sample can be used to check that.
The width of the confidence intervals was very small which meant my sample value for the mean was close to that of the real parent population. Both the adult and child text had small intervals (0.6 and 0.5) suggesting that the sample taken was actually very accurate.
The width of the confidence interval increases as the confidence percentage increases. This can be clearly seen from the calculations that were done with the 95% and 90% confidence intervals.
Overall the confidence intervals have proven my hypothesis, but also shown how detailed statistic process can be used to attain and confirm different unknown measures that need to be found and proven
Confidence intervals limitation
For example if I wanted to be 95% sure that the sample mean was within 0.1 of a letter of the population mean for both texts I would need to take X number of samples. The smaller the sample the more accurate the investigation is:
Adult Text
Child Text
Limitation
One major limitation was the amount of samples that I collected. If I had collected more samples my data would have increased in accuracy. Collecting 50 samples from both the books seemed sensible as this was sufficient for results analysis and also co-agreed with the central limit theorem. If I were to repeat the investigation I would increase the number of samples that I collected because this would increase the accuracy of the investigation in generating a more accurate set of results that give a more detailed portrayal of the real parent population.
Extension
To extend the investigation I could have looked at the number of words per line. I could have also looked at the number of paragraphs per page. The investigation can also be extended so that a much larger sample of the current texts are taken i.e. 500 and from the results the confidence intervals could show how accurate my set of means from this sample were in order to determine the reliability of this investigation
So if I was to extend this investigation I would firstly take more samples to ensure greater accuracy which would therefore allow greater certainty in any conclusions drawn. Secondly I would compare a number of different horror books against each other to see if their population parameters were similar or if they varied. Another progression could be to sample a number of horror books by the same author to see if they are at all similar in their population parameters.
If I was to do this coursework again I could change the populations. I could possibly choose a book which has an adult's version and a child's version. I feel this would be fairer because it is a possibility that I chose an advanced child's book or an easy adult's book. Instead of collecting two sets of data for each population I could have chose two adults and two child's books. I then could have taken 100 words from each book and combined the child's and the adults together. Another possible way would be to choose two fact books. It would be interesting to see if this had any difference on the final results.
Improvements
To improve the investigation, I could have collected more results. This would lead to the sample mean being more similar to the population mean.
I could have also collected different types of results. I could have looked at the number of words per page.
Conclusion
In conclusion, my results show that it is possible to gain information about authorship of a text using statistical measures. My results show this because the adult text has a higher average of letters per word and also has more variation of word length. However more information to check the reliability of the investigation can be found by extensive use of larger samples.
Overall the investigation was a success as my hypothesis were proved right and the set of data seemed to be accurate but the reliability can only be tested on grounds that the real mean of the parent population is known and from this the consistency of this investigation can be ensured. My sampling method and generating of random numbers was also good so I would change the method, instead I would opt use a much larger size sample
Khaled Hamid Page
