Mathematics Applications and Interpretation Internal Assessment

To what extent does age affect the reaction time?

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Table of Contents:

INTRODUCTION 3
RATIONALE AND AIM 3-4

METHODOLOGY 4-5

HYPOTHESIS 5

CALCULATIONS
FINDING AVERAGES 5-6
PEARSON'S PRODUCT MOMENT CORRELATION COEFFICIENT 7-10
SPEARMAN'S RANK CORRELATION COEFFICIENT 11-13

ANALYSIS 6-13

CONCLUSION 13-14

BIBLIOGRAPHY 14

APPENDIX 15-16

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Introduction:

Rationale and aim

I have chosen to do an investigation on age and reaction times because such situations may

arrive in my chosen field of career, medicine. When new medicines go through trials, many

tests are done to the test subjects. With the use of correlation, conclusions like how fast a

specific amount of dosage can treat the patient can be reached. Hence why knowing how to

conduct a correlation investigation is significant. I have studied about correlation and the

reaction times of humans in my GCSE course of biology so I have the knowledge of the

different variables that can affect reaction times in humans.

The aim of this investigation is to obtain how much impact does the age of humans affect their

reaction rates. I will be determining the results of this investigation by conducting a scatter plot

of the data, if the scatter graph shows linear correlation I will use Pearson’s product moment

correlation coefficient, and if the scatter graph shows non-linear correlation, I will use

Spearman’s rank correlation coefficient. And the value of the correlation will quantify the

strengths of the correlation.

Methodology:

It is a known fact that the older a person gets, the slower their reaction time. I did not do this

experiment with people aged between 19 and 23 because after the teenager phase, the brain’s

response time starts to decline at the age of 24 which would be an essential part to the data that

has been collected1 and be considered as an anomaly. With reactions from the age of 24, a

meaningful change in reaction times will be seen between teenagers and adults. To arrive at a

1
https://www.pbs.org/newshour/science/brains-reaction-time-peaks-age-24-study-finds

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valid and reliable conclusion, I have done the reaction test with 7 people from each age group

(5 times each) making sure they have not consumed any caffeine before the experiment. I have

chosen caffeine as one of the major control variables because it can slow down reaction times

and this would have led to unreliable results. To conduct this experiment, I kept all 21 of my

candidates’ diet caffeine free at least two days prior to the experiment so that the data would

be valid. When I had gone to India during the summer holidays, I had the conducted this

reaction time experiment on some family members and then some friends and family back in

the United Arab Emirates. The raw data of each child/teenager/adult’s reaction time each time

they did it is in the appendix (table 5, table 6, and table 7).

My independent variable for this experiment is the different age groups and the dependent

variable as the number of seconds it took them to catch the ruler. I then converted the data in

seconds to milliseconds to be able to generate a more visible graph on Microsoft excel. My

other control variables were keeping the distance between the hand of the candidate and the

long stick the same (5cm).

Once I made sure all my control variables were controlled, I held the long stick 5 centimeters

away from the hand which would catch it and as soon as I would drop the long stick, a third

person will start the timer and stop when the candidate would catch it. Similarly, I repeated this

5 times with each candidate (raw data in the appendix). The way I conducted the experiment is

illustrated in figure 22.

2
https://www.biologycorner.com/worksheets/response_time.html

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Hypothesis:

Due to my previous knowledge from Biology, my null hypothesis (H0) for this investigation

would be that reaction times are independent of age and my alternate hypothesis (H1) would be

that reaction times are dependent of age.

Calculations:

Finding Averages

To plot the scatter graph of the mean values, I had to first calculate the average for each person.

To find the average, I added all 5 reaction times and then divided the total value by 5.

For example:

Table 1: Table showing the 5 reaction times for 12-year-old child

Age Reaction time of children (milliseconds)

12 230 200 170 220 180

To find the average of the reaction time of the first child:

230+200+170+220+180
Average = 5

Average = 1000/5 = 200 milliseconds

Similarly, I calculated each person’s averages and put it into a table.

Analysis:

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 Table 2: Table showing the mean reaction times of 7 children (6-12):

Mean reaction time

Age (milliseconds)

200
12

180
11

160
11

180
12

160
8

180
10

160
6

Graph 1: Graph showing the mean reaction times of 7 children (6-12):

Mean reaction times of 7 children (6-12)

250
Mean reaction time (milliseconds)

200

150

100

50

0
0 2 4 6 8 10 12 14
Age

With this graph it can be claimed that as the age increased, the mean reaction time of the

children also increased. For age 11, it is visible that there are two different mean values – 160

and 180 and age 12 has two different values as well – 180 and 200. The reason why the values

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could be different for the same age is because as individuals both children are different.

Children in the 180-200 milliseconds range could be more active with their feet which could’ve

had a major impact on their reaction time with their hands. These environmental factors cannot

be controlled and hence the difference of reaction time with other children.

Calculating Pearson’s Product Moment Correlation Coefficient

Since this is a linear graph, I will be conducting Pearson’s product moment correlation

coefficient. The formula to calculate Pearson’s PMCC3:

𝑠𝑥𝑦
r= , where:
√𝑆𝑥 𝑆𝑦

Sxy = n (∑ 𝑥𝑦) − (∑ 𝑥) (∑ 𝑦);

Sx = [𝑛 ∑ 𝑥 2 − (∑ 𝑥)2 ] and;

Sy = [𝑛 ∑ 𝑦 2 − (∑ 𝑦)2 ]

𝑛 (∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦)
r=
√[𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 ][𝑛 ∑ 𝑦 2 −(∑ 𝑦)2 ]

In this case: n is the number of trials, x is the children’s age, and y is the mean reaction time.

∑ 𝑥 = 11 + 12 + 12 + 11 + 8 + 10 + 6 = 70

∑ 𝑦 = 200 + 180 + 160 + 180 + 160 + 180 + 160 = 1220

∑ 𝑥 2 = 122 + 112 + 112 + 122 + 82 + 102 + 62 = 730

∑ 𝑦 2 = 2002 + 1802 + 1602 + 1802 + 1602 + 1802 + 1602 = 214000

∑ 𝑥𝑦 = (200 × 12) + (180 × 11) + (160 × 11) + (180 × 12) + (160 × 8) + (180 ×

10) + (160 × 6) = 12340

7 (12340)−(70)(1220)
r=
√[7(730)−(70)2 ][7(214000)−(1220)2 ]

980
r= = 0.690
√2016000

3
https://www.youtube.com/watch?v=zBAOPENjRBc&t=182s

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This PMCC value for children between the age of 6 to 12 was a moderately positive correlation.

The reason for the correlation not being strong is because of some of the uncontrollable factors

in this investigation. Children are usually very involved in many different activities during their

childhood and even genetics sometimes play a role in reaction times, for example ADHD4

(attention deficit hyperactivity disorder) which is one of the most common neurodevelopmental

disorders of childhood5.

Next, I calculated the mean reaction time of teenagers age from 13 to 18, generated a graph off

Microsoft excel and then calculated Pearson’s PMCC.

Table 3: Table showing the mean reaction times of 7 teenagers (13-18):

Mean reaction time

Age (milliseconds)

250
18

240
18

250
17

210
15

200
14

190
13

230
17

4
https://www.xcode.life/genes-and-personality/how-genes-influence-your-reaction-
time/#:~:text=Reaction%20time%20is%20influenced%20in,its%20effects%20on%20reaction%20time.
5
https://www.cdc.gov/ncbddd/adhd/facts.html#:~:text=ADHD%20is%20one%20of%20the,and%20often%20la
sts%20into%20adulthood.

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 Graph 2: Graph showing the mean reaction times of 7 teenagers (13-18):

Graph showing the mean reaction times of 7 teenagers (13-18)

300
Mean reaction time (milliseconds)
250

200

150

100

50

0
0 5 10 15 20
Age

Looking at this graph, it can be stated that the older the teenager the greater the mean reaction

time. In the previous graph (graph 1) there were two different values for the same age and even

in this graph (graph 2) the data for teenagers shows the same. Children and teenagers go

through a growing phase which ends up affecting their reaction times leading to many

similarities between them. The data of children and teenagers don’t have a huge gap between

them unlike a huge gap between teenagers and adults as presented later in the exploration.

In this investigation, I also got to know that when young tweens/teenagers hit puberty, their

reaction time improves, hence which proves the small difference in reaction times between

children and young teenagers (13-14). At age 13 – around the puberty hitting age – the reaction

time is the lowest and from the previous data one of the 12-year-old had a reaction time of 200

milliseconds proving that during puberty reaction time can improve. Females tend to level off

from 15 to 17 years6 which makes sense with the graph presented. The teenager aged 14 is a

female and so is the 17-year-old with the reaction time of 250 milliseconds. The data supports

6
https://www.tandfonline.com/doi/abs/10.1080/10671315.1975.10616684#:~:text=Mean%20reaction%20tim
es%20decreased%20markedly,and%20males%20were%20consistently%20faster.

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the point made by the authors Fulton and Hubbard; the reaction time of the 17-year-old is

obviously greater than the 14-year-old establishing two things: reaction times do slow down

after puberty for girls and that my alternate hypothesis for this investigation is true which is

that reaction times are dependent of age. The gender also plays a role in the reaction times;

however, I haven’t focused on specific gender instead I have looked at different age groups.

This has had an impact on my results due to gender also being an important control variable.

I calculated Pearson’s PMCC for teenagers as well because the line on the graph shows a linear

correlation. The PMCC value for teenagers is 0.956 which shows a strongly positive

correlation. A gradual increase in reaction times from children to teenagers shows that

increasing age does in fact increase reaction times regardless of some different values from the

raw data because they could’ve been simply affected by the uncontrollable environmental and

genetic factors.

Lastly, I calculated the mean reaction time of adults, generated a graph off Microsoft excel and

then calculated the Spearman’s rank correlation coefficient.

Table 4: Table showing the mean reaction times of 7 adults (24-48):

Mean reaction time

Age (milliseconds)

39 440

36 410

24 350

48 500

26 330

31 370

28 340

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 Graph 3: Graph showing the mean reaction times of 7 adults (24-48):

Graph showing the mean reaction times of 7 adults (24-48)

Mean reaction time (milliseconds) 600

500

400

300

200

100

0
0 10 20 30 40 50 60
Age

In this graph, there is a positive relationship between the mean reaction time and age, just like

the previous graphs. It can be seen from the graph that the oldest adult has the highest mean

reaction time, nevertheless, the youngest adult does not have the lowest mean reaction time.

As I previously mentioned this could have been due any of the uncontrollable environmental

and genetic factors. Ignoring the 24-year-old’s reaction time, there is a strong linear line

presented on the graph.

To determine whether the correlation is strong, moderate, or weak, I used the Spearman’s rank

correlation coefficient as the scatter shows a curved relation. Pearson’s PMCC and Spearman’s

rank correlation coefficient was used to tell how strong the correlation between two variables7.

If the degree of scatter (denoted as r) is a positive number (it will be closer to 1), then the

correlation is strong positive and if the r value is a negative number (closer to -1), then the

correlation is strong negative.

7
https://revisionmaths.com/advanced-level-maths-revision/statistics/product-moment-correlation-coefficient

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Calculating Spearman’s Rank Correlation Coefficient

I have conducted Spearman’s rank correlation coefficient8 below.

Table 4: Table showing the ranking of adults’ age and their mean reaction time

Number of Age of Rank Mean reaction time of Rank Difference Difference

adults adult adult (milliseconds) squared
1 39 2 440 2 0 0
2 36 3 410 3 0 0
3 24 7 350 5 -2 4
4 48 1 500 1 0 0
5 26 6 330 7 1 1
6 31 4 370 4 0 0
7 28 5 340 6 1 1
To calculate Spearman’s rank correlation coefficient, the formula used is:

6 ∑ 𝑑2
𝜌 = 1 − 𝑛(𝑛2𝑖 −1)

In this case: n is the number of adults and d2 is the rank difference squared.

∑𝑖 𝑑 2 = 0 + 0 + 4 + 0 + 1 + 0 + 1 = 6

6×6
𝜌 = 1 − 7(72−1)

36
𝜌 = 1 − 336 = 0.893

This Spearman’s rank correlation coefficient value illustrates that there is a strong positive

correlation between the mean reaction times and the age of the adults.

Conclusion:

After investigating my research question: to what extent does age (6-12, 13-18, and 24-48

years) affect the reaction time, I have concluded that age affects reaction time to a great extent.

8
https://www.youtube.com/watch?v=XGRqcI2R0H8

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Even though other factors also play a role in reaction times, age is clearly one of the very

significant factors. For example, in teenagers it was mentioned that puberty can lower reaction

times and puberty arrives with increasing age, therefore proving the significance of age. As an

individual grows older, the chances of getting genetic diseases or stress-related diseases also

increases. Most of the environmental and genetic factors which were uncontrollable for this

investigation are also impacted by age. The aim of my investigation was to obtain how much

impact age has on reaction times and as I had predicted earlier, the reaction time does get slower

as age increases. There were two strongly positive correlations and one moderately positive

which then leads to the rejection of the null hypothesis I had made earlier in the investigation.

The results of this investigation will definitely help me in my future decisions as a doctor which

makes me glad for choosing this informative inquiry to conduct.

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 Bibliography:

“The product moment correlation coefficient” Revision Maths

https://revisionmaths.com/advanced-level-maths-revision/statistics/product-moment-

correlation-coefficient

Shalby, Colleen. “Your brain’s reaction peaks at age 24, study finds” PBS news hour

https://www.pbs.org/newshour/science/brains-reaction-time-peaks-age-24-study-finds

“The product moment correlation coefficient” Revision Maths

https://revisionmaths.com/advanced-level-maths-revision/statistics/product-moment-

correlation-coefficient

“Using the scientific method to measure multitasking and response time” Biology corner

https://www.biologycorner.com/worksheets/response_time.html

Essa. “How to find the Pearson’s correlation coefficient (by hand) Prof. Essa

https://www.youtube.com/watch?v=zBAOPENjRBc&t=182s

Fulton, Clifton and Alfred W. Hubbard. “Effect of puberty on reaction and movement times”

Taylor & Francis online

https://www.tandfonline.com/doi/abs/10.1080/10671315.1975.10616684#:~:text=Mean%20r

eaction%20times%20decreased%20markedly,and%20males%20were%20consistently%20fas

ter.

Essa. “How to find the Pearson’s correlation coefficient (by hand) Prof. Essa

https://www.youtube.com/watch?v=XGRqcI2R0H8

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 Appendix:

Raw data: table 5 showing reaction time of children

Age of Reaction time of children (milliseconds)

Children Trial 1 Trial 2 Trial 3 Trial 4 Trial 5

12 230 200 170 220 180

11 250 220 210 110 130

11 190 160 140 140 180

12 250 130 190 210 140

8 190 170 150 170 130

10 240 180 140 180 150

6 230 180 190 120 90

Raw data: table 6 showing reaction time of teenagers

Age of Reaction time of teenagers (milliseconds)

Teenagers Trial 1 Trial 2 Trial 3 Trial 4 Trial 5

18 260 220 270 260 250

18 250 200 240 220 270

17 250 260 240 250 240

16 210 230 200 190 200

14 190 230 220 180 200

13 170 190 180 200 230

17 210 180 240 270 230

Raw data: table 7 showing reaction time of adults

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Age of Reaction time of adults (milliseconds)

Adults Trial 1 Trial 2 Trial 3 Trial 4 Trial 5

39 430 420 450 440 450

36 400 420 390 420 410

24 340 340 350 340 360

48 510 500 510 480 490

26 330 320 340 330 320

31 380 390 350 360 350

28 340 360 330 340 350

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