Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 3, No. 2, 2023 

 

39 

Analysis and Prediction of Wordle Dataset Based on 
ARIMA and Pearson's Correlation Coefficient 

Zixin Wan 

School of Computer and Information Engineering, Hubei University, Wuhan 430062, China 

3064253289@qq.com 

 

Abstract: In this paper, a time series analysis was performed to ensure the parameters p, d, and q. Finally, the ARIMA (1,1,0) 

model was chosen to create a prediction interval for the number of outcomes to be reported in the future. The results of the 

precision test showed good confidence in the prediction with a goodness-of-fit of 98.2%. the prediction interval for March 1, 

2023 was [10103,10523]. In addition, this paper used Pearson's correlation coefficient and chi-square test to verify the correlation 

between each attribute of the word and the percentage of scores in the hard model. In summary, this paper found that the attributes 

of words definitely affect the percentage of scores, and these scores are mainly the initials and lexicality of words. 

Keywords: ARIMA; Pearson's correlation coefficient; Chi-square test. 

 

1. Introduction 

In recent years, Wordle, a word-guessing game with a 

matrix of yellow and green squares as an interactive interface, 

has become popular on social networks as people become 

more aware of innovation and prefer smart games [1]. Text 

analysis refers to the representation of text and the selection 

of its feature terms; text analysis is a fundamental problem in 

text mining and information retrieval, which quantifies the 

feature words extracted from text to represent text 

information.By examining Wordle dataset, this paper is able 

to predict the number of reports for Wordle operators and 

provide assistance with their word selection[2]. 

In this paper ARIMA is used, a more refined and accurate 

algorithm for analyzing and forecasting time series data is the 

Box-Jenkins method, whose common models include: 

autoregressive model (AR model), sliding average model 

(MA model), (autoregressive-sliding average hybrid model) 

ARMA model, and (differentially integrated moving average 

autoregressive model) ARIMA model. We consider building 

ARIMA model to make prediction on the Wordle Dataset, 

taking into account certain errors to obtain the predicted 

results. 

2. Acquisition of Data and 
Assumptions 

The data used in this paper are taken from problem C of the 

2023 American College Mathematics Modeling Contest. To 

facilitate the analysis of the problem, the data used in this 

paper make the following assumptions: (1) The data provided 

are mostly correct and have negligible impact on the accuracy 

of the analysis of the results [3]. (2) There are no external 

factors to motivate users to play the game, and growth and 

decline in data is common. (3) There are no users sliding 

scores or making scores or making intentional mistakes. 

3. Data preprocessing 

The Wordle datasets gives the word statistics from January 

7, 2022 to December 31, 2022, a total of 358 days. Since we 

are only allowed to use the data file ‘Problem C Data 

Wordle.xlsx’ provided by COMAP official, we need to pre-

process the data before solving the problem. 

To facilitate our later work and to ensure the reliability and 

plausibility of the result, data pre-processing was conducted 

on the datasets as follows. 

3.1. Missing Value Processing 

We first checked the whole table for missing values using 

Python and found that there are no missing values in 'Problem 

C Data Wordle.xlsx'. 

3.2. Outlier Detection 

In the process of outlier detection, we handle the daily 

words and data respectively to eliminate the abnormal ones. 

3.2.1. Word processing 

According to the rules of the game, the word length can 

only be 5, so we need to remove the data of words whose 

length is not 5. First of all, we need to detect whether there 

are spaces and punctuation marks in the words, and this step 

removes the spaces in the word FAVOR; then the words 

whose length is not 5 are removed, and this step removes the 

words CLEN and TASH. 

According to the rules of the game, words can only be 

composed of 26 English letters, so it is necessary to detect 

foreign words that are not composed of English letters, and 

this step removes the word naïve. 

3.2.2. Numerical processing 

(1) Removal of outliers 

We plotted the scatter plot based on the column Number of 

reported results and found that the word STUDY in number 

529 was an outlier, which was not consistent with the 

statistical trend of the number in the last month, so we 

removed it [4]. 

(2) Calculating percentages 

Considering that the percentages of players solving the 

puzzle are likely to be rounded and that the attempt limit 

should be no more than 7, therefore the calculating sum 

interval should be between 97 and 103, either all discarded or 

all reserved. For this reason, we find that the percentages of 

the word nymph sum to 126, which is obviously beyond the 

error interval [5]. 

(3) Reversing the date 



 

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The dates given in the question are from December 31, 

2022 to January 7, 2022. To facilitate the later tasks such as 

predictions, we arrange the data in positive date order in this 

pre-processing section. 

 

Figure 1. The scatter of Number of reported results and day that 
help pick out the outlier 

4. The Interval Prediction Model Based 
on ARIMA 

4.1. Time Series Analysis 

The data to be analyzed in this task changes with time 

constantly, and our task is to predict the trend of data 

development over time, so we use a time series model to 

complete the job.  

By means of time series analysis, we can achieve the 

purpose of describing the past, analyzing the regularity, and 

predicting the future. Also, we can use this as an access point 

to predict the number of reports in the future period [6]. 

4.2. Stationarity test 

Before using a time series model, we have to test the data 

for stability to determine whether the data set can be used in 

a time series model; stability requires that the fitted curve 

obtained from the sample time series can continue inertially 

in the future period with the existing pattern. 

(1) Time-series test diagram  

 

Figure 2. Time-series test diagram 

From the above time series plot, it can be found that the 

trend from January to February 2022 shows an upward trend, 

reaching the highest point on February 2, and from February 

to December is a gradually stabilizing downward trend, and 

from July 2022, the curve can be fitted with the Sigmod 

function after translation.  

Therefore, based on the trend of the time series graph, the 

data set can be predicted using a time series model from a 

subjective perspective. 

(2) Unit-root test 

The unit-root test is to test whether there is a unit root in 

the series and whether the presence of a unit root is a non-

stationary time series. The most frequently used one is the 

ADF test. 

Hypothesis of ADF test: We first assume the existence of a 

unit root and then perform a significance test. If the test 

statistic P is less than 10% (5% or 1%) confidence level, then 

we believe that the original hypothesis is rejected with 90% 

(95% or 99%) confidence, that is, there is no unit root and the 

series is smooth and stable [7]. 

Table 1. ADF test 

Difference order P 

0 0.003*** 

1 0.001*** 

Note: *** stands for 1% significance level 

The results of this serial test show that based on the variable 

Number of reported results, the significance P-value is 0.003 

at the difference of order zero, which is less than 0.01 and 

presents significance at the level, indicating that the original 

series is a stationary sequence; at the difference of order one, 

the significance P-value is 0.001, which is also less than 0.01, 

indicating that the time series after the first order is also a 

stationary sequence with better stability[8].  

In contrast, we choose the first-order difference, that is, the 

parameter "d" is 1, and initially determine the ARIMA model 

as ARIMA (p,1, q). 

4.3. White noise test 

After the stationarity test, we also have to do the white 

noise test, which is to test whether the series belongs to the 

pure random series. This is because if the series is completely 

random, the past behavior will not have any influence on 

future development, and the subsequent study will be 

meaningless [9]. According to the results of the stationarity 

test, the original time series of the Number of resulted reports 

is stable, so we perform a white noise test on the original 

series by lagging backward 40, and plot the range of p-value 

for the white noise test below. 

 

Figure 3. Pure Randomness Testing 

As can be seen from the figure, the p-values of the 

randomness tests for the number of resulted reports are all 

significantly less than 0.05, indicating that the sequence is 

non-random, which provides a basis for subsequent 

prediction. 

4.4. ARIMA Model based on Time Series 

Prediction  

(1) Selection of p and q 

By means of the unit root test, we determined that the series 

is a smooth series after the first order difference. Here, we 

make the autocorrelation and partial autocorrelation plots of 

this stationary series using python as follows. 



 

41 

 

(a)The autocorrelation 

 

(b) the partial autocorrelation. 

Figure 4. ACF and PACF  

From Figure 4(a), we can see that the series is truncated at 

order zero with a preliminary judgment of q=0. From Figure 

4(b), the series is truncated at order 1 with a preliminary 

judgment of p=1.Combining the above, we initially 

determined the ARIMA model as ARIMA (1,1,0). 

(2) Prediction 

Based on the model ARIMA (1,1,0), we forecast the 

number of resulted reports on March 1, 2023, and the fitted 

forecast graph is as follows. 

 

Figure 5. The prediction of the number of resulted report base on 
ARIMA 

The above figure represents the original data graph, model 

fitted values, and model predicted values for the number of 

resulted report series. From the figure, it can be visually seen 

that the fitted curve overlaps well with the original data curve, 

and the changing trend is basically the same, which indicates 

that the model we used fitted very well. We concluded that the 

number of reported results on March 1, 2023 predicted by the 

ARIMA (1,1,0) model was [10103,10523] with a precision of 

98.2%. 

5. Attribute Extraction 

5.1. Parts of Speech 

Part of speech is one of the top features of words. Given 

that this attribute may affect the correctness of users' guesses, 

here we apply Python's natural language processing NLTK 

library to get the part of speech of each word. Our result 

statistics show that the number of nouns is predominant, 

followed by adjectives, so we assume that the percentage of 

scores in hard mode with the same property can be correlated 

[10]. 

5.2. Word Initials 

According to the number of words counted by the Oxford 

English Dictionary, the number of words beginning with 

different letters possess some characteristics, for example, the 

number of words beginning with the letters s and p is the 

largest while letters beginning with x and y has the smallest 

proportion. In light of this, we presume that the initial letters 

can have a certain influence on the correct rate of vocabulary 

guessing in Wordle. The following is the frequency graph of 

the initial letters of words after pre-processing the data using 

Python. 

 

(a)The frequency graph of initial letters 

 

(b)The sum of letter pairs 

Figure 6. The frequency graph of initial letters and the sum of 
letter pairs 

The result that the largest number of words beginning with 

the letter s and the smallest number of words beginning with 

the letters x, y, and z were obtained, which were 

approximately the same as the regular patterns in the Oxford 

English Dictionary. Therefore, the prediction of future words 

based on the statistical quantities of the whole has strong 

reliability to some extent. 

5.3. Adjacent Letters Within a Word 

In general, the composition of each word is a sequence of 

adjacent letters generated by phonetic symbols, take the word 

track for instance, pronounced /træk/, can be decomposed into 



 

42 

/tr/ and /k/ two common phonetic symbols, the letter pair of 

/tr/ is tr, /k/ has ck, ke and other letter pairs. From this, we 

numbered the word letters 1~5 from left to right, then match 

them adjacently to form a letter pair and count the total 

number of occurrences of 6 letter pairs in each word in all 

words. This is used to indicate the commonness of the word 

and judge its correlation with the percentage of difficulty 

pattern scores. 

From Figure 6 (b), we found that the sum of letter pairs for 

each word was unstable, and speculate that this may have 

some relationship with the percentage of difficulty patterns. 

5.4. Number of Identical Letters in Each Word 

If two or more letters are identical in a word, the total 

number of guesses required will decrease to some extent. That 

is, the number of repetitions of letters in a word affects the 

level of difficulty of the question to a greater or lesser extent. 

Thus, we extracted the number of repetitions of the letters of 

all the words in the processed datasets. The result was that 

there were 254 words with different letters, 98 words with two 

identical letters, 2 words with three repetitive letters, and no 

words with four or more identical letters. 

5.5. The Ratio of Vowel Letters to Consonant 

Letters 

Consonants and vowels depend on each other to form the 

diverse syllables of words. After reading related articles, we 

found that people prefer to use words with more vowels, so 

we calculated the ratio of vowel letters to consonant letters 

and guessed that the number of vowel letters would be related 

to the percentage of scores in the difficult mode. 

6. Correlation Analysis 

6.1. Method Selection 

For this part, we have to determine the relationship between 

the founded attributes (X) and the score played in hard mode 

(Y). 

The dependent variable Y is quantitative in this question, 

while the independent variable X is composed of different 

types when representing different attributes. However, for the 

two attributes of part of speech and word initials, the 

relationship between them belongs to quantitative and 

categorial, so we cannot use the correlation coefficient 

directly. We expressed the quantitative representation of 

fractional percentages as multiple interval fixed-class 

representations and used the chi-square test to determine the 

relationship between them. 

6.2. Correlation coefficient 

Before choosing the correlation coefficient, we need to 

determine whether the data satisfy normality, and use Pearson 

correlation coefficient if it satisfies, or Spearman correlation 

coefficient if it does not. 

The difference between Spearman correlation coefficients 

(SCC) and Pearson correlation coefficients (PCC) is that SCC 

takes the correlation coefficient as a ranking variable, so we 

first need to calculate the correlation coefficient r   by the 

following formula: 

( )( )
1

2 2

1 1

r

( ) ( )

n

n

i i

i i

i

n

i i

X X Y Y

X X Y Y

=

= =

− −

=

− −



 
         (1) 

Here iX
 (i=1,2,...,354) represents the independent 

variable, meaning the attributes of the word, and iY

(i=1,2,...,354) denotes the dependent variable, meaning the 

scores in hard mode, and X and Y denote the mean of the 

two, respectively. 

The data satisfying the normal distribution can be judged 

by calculating the correlation coefficient, and the closer the 

correlation coefficient is to 1, the stronger the correlation is; 

for the data not satisfying the correlation coefficient, we have 

to grade them after calculating the correlation coefficient, and 

then calculate it by the following formula: 
2

2

6
1

( 1)

id

n n
 = −

−


                (2) 

Here d denotes the descending position and n denotes the 

number of samples. The correlation between them is then 

determined by calculating out of ρ. 

 

(a) 

 

(b) 

 

(c) 

Figure7. (a) - (c) plot the frequency of 3 attributes, the number of 
the same letter, the sum of letter pairs in a word and the ratio of the 

number of vowels to consonants, respectively. 



 

43 

The number of same letters, the sum of letter pairs and the 

ratio of the num of vowels to consonants all basically satisfies 

the normal distribution, so for this attribute, we use the 

Pearson correlation coefficient, and calculate from the above 

formula are 0.086, 0.089 and 0.046 respectively, indicating 

that the correlation between the two isn't apparent. 

6.3. Chi-square test 

Before conducting the chi-square test we first classified the 

percentages of difficult mode scores (Y) into intervals, then 

counted the number of different categories of attributes in 

each interval, and finally conducted the chi-square test. We 

divided the data set of percentages of scores in hard mode 

evenly into four intervals as follows. 

Table 2. Four intervals 

0.012,0.063 0.063,0.083 0.063,0.083 0.093,0.133 

For the two attributes of word initials and part of speech, 

the reliability of the hypothesis is first calculated by drawing 

separate frequency lists and assuming that they are correlated, 

using the following formula: 
2

2

1

( )i i

i

k

i

f np
x

np=

−
=               (3) 

in which x (i=1,2, ...) denotes the group frequency, which 

is the true value in the table, n  denotes the total number of 

samples, and ip
  denotes the probability of occurrence of 

each sample. The greater the value of the hypothesis, the 

greater the reliability of the hypothesis and the correlation. 

The correlation reliability can be determined according to the 

following table. 

Table 3. The precision of chi-sequare test 

( )kP 2  0.05 0.025 0.010 0.005 0.001 

k 3.841 5.024 6.635 7.879 10.828 

(1) Initials words 

Table 4. The number of different initials in different interval 

interval a b c ... sum 

[0.012,0.063) 5 3 10 ... 87 

[0.063,0.083) 8 7 7 ... 83 

[0.083,0.093) 7 5 8 ... 94 

[0.093,0.133] 8 5 7 ... 89 

sum 28 20 32 ... 354 

Using the above formula and the table, we calculate the 

value of x as 4.889 and indicate that correlation probability 

lies between 0.95 and 0.9725, so it can be determined that the 

word initials and the score in hard mode are highly correlated. 

(2) Part of speech 

Similarly, the value of x calculated here is 4.123, and the 

result is that there’s a strong correlation in the attribute of part 

of speech. 

Table 5. The number of parts of speech in different interval 

c NN JJ MD ... sum 

[0.012,0.063) 44 24 2 ... 87 

[0.063,0.083) 63 15 0 ... 83 

[0.083,0.093) 45 23 0 ... 94 

[0.093,0.133] 53 23 0 ... 89 

sum 205 85 2 ... 354 

7. Conclusion 

In this paper, we perform time series forecasting and 

correlation analysis. To obtain more accurate results, this 

paper selects an effective model and continuously optimizes 

the parameters. Finally, we obtained the ARIMA (1,1,0) 

model with a fit of 98.2%, which shows that the model has a 

good predictive power. In addition, five attributes were 

derived and correlation analysis was performed between 

attributes and percentages in the hard model. It was found that 

the attributes of the words definitely affect the percentages of 

the scores, which are mainly the initials and the lexicality of 

the words. 

References 

[1] Zeng B, Tong M, Ma X. A new-structure grey Verhulst model: 
development and performance comparison[J]. Applied 
Mathematical Modelling, 2020, 81: 522-537. 

[2] Wang Z, Dang Y, Liu S. Unbiased grey Verhulst model and its 
application[J]. Systems Engineering-Theory & Practice, 2009, 
29(10): 138-144. 

[3] Ariyo A A, Adewumi A O, Ayo C K. Stock price prediction 
using the ARIMA model[C]//2014 UKSim-AMSS 16th 
international conference on computer modelling and 
simulation. IEEE, 2014: 106-112. 

[4] Wang Z X, Li Q. Modelling the nonlinear relationship between 
CO2 emissions and economic growth using a PSO algorithm-
based grey Verhulst model[J]. Journal of Cleaner Production, 
2019, 207: 214-224. 

[5] Zeng B, Ma X, Zhou M. A new-structure grey Verhulst model 
for China’s tight gas production forecasting[J]. Applied Soft 
Computing, 2020, 96: 106600. 

[6] Tang L, Lu Y. Study of the grey Verhulst model based on the 
weighted least square method[J]. Physica A: Statistical 
Mechanics and its Applications, 2020, 545: 123615. 

[7] Hillmer S C, Tiao G C. An ARIMA-model-based approach to 
seasonal adjustment[J]. Journal of the American Statistical 
Association, 1982, 77(377): 63-70. 

[8] Sowell F. Modeling long-run behavior with the fractional 
ARIMA model[J]. Journal of monetary economics, 1992, 29(2): 
277-302. 

[9] Fattah J, Ezzine L, Aman Z, et al. Forecasting of demand using 
ARIMA model[J]. International Journal of Engineering 
Business Management, 2018, 10: 1847979018808673. 

[10] Nochai R, Nochai T. ARIMA model for forecasting oil palm 
price[C]//Proceedings of the 2nd IMT-GT Regional 
Conference on Mathematics, Statistics and applications. 2006: 
13-15.