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The Correlation Between Music and Math: A Neurobiology Perspective
Biology 202
2002 First Paper
On Serendip
The Correlation Between Music and Math: A Neurobiology Perspective
Cindy Zhan
I remember the first time I heard the statement "Did you know that
listening to classical music enhances your mathematical abilities?" I
was both intrigued and excited, intrigued because I did not understand
how music and math, two seemingly unrelated subject could possibly
affect each other. I was also excited because I began to view classical
music as some kind of magical potion that would transform my math
skills from decent to extraordinary. When I had the opportunity to
write this web paper, I immediately jumped into the topic of music and
math. The questions that I wish to answer throughout this paper are;
does listening to music really help you do better in math? If so, which
part of the brain is controlling the correlation between math and
music? In addition, how does music stimulate the brain in a way that
enhances mathematical abilities?
It turns out that there is much evidence that supports the positive effects of music on one's ability to do math. Most research shows that when children are trained in music at a young age, they tend to improve in their math skills. The surprising thing in this research is not that music as a whole is enhancing math skills. It is certain aspects of music that are affecting mathematics ability in a big way. Studies done mostly in children of young age show that their academic performance increases after a certain period of music education and training. One particular study published in the journal 'Nature' showed that when groups of first graders were given music instruction that emphasized sequential skill development and musical games involving rhythmn and pitch, after six months, the students scored significantly better in math than students in groups that received traditional music instruction. (1)
The result of this study posed another important question. How does
this type of music that emphasized sequential skills, rhythmn and pitch
manage to improve children's ability to do math? It turned out that
there are two distinguished types of reasoning, spatial temporal (ST)
reasoning and Language analytical (LA) reasoning. LA reasoning would be
involved in solving equations and obtaining a quantitative result. ST
reasoning would be is utilized in activities like chess when one needs
to think ahead several moves. The effect of music on math sometimes
termed the Mozart effect. The Mozart effect gain its name after the
discovery that listening to Mozart's compositions, which is very
sequential, produces a short-termed enhancement of spatial-temporal
reasoning. Some key reasoning features used in spatial temporal
reasoning are
1. The transformation and relating of mental images in space and time
2. Symmetries of the inherent cortical firing patterns used to compare physical and mental images and
3. Natural temporal sequences of those inherent cortical patterns (3).
The same people who conducted the Mozart effect experiment also
suggested that spatial-temporal reasoning is crucial in math. The areas
of math that require ST reasoning are geometry and certain aspects of
calculus, which require transformations of images in space and time. In
higher mathematics, the ability to write mathematical proofs is also
associated with ST reasoning because proof writing is a task that
requires intuitive sense of natural sequences and the ability to think
ahead several steps.
As to the question, what part of the brain controls the correlation between math and music, there are also many resources that provide answers. Dr. Gottfried Schlaug, found that certain regions of the brain such as the corpus callosum and the right motor cortex, were larger in musician who started their musical training before the age of 7 (2). As to what happens in that area of the brain when one listens to music, we turn to the experiment performed by Xiaodeng Leng and Gordon Shaw. Gordon and Leng developed a model of higher brain function, which is based on the trion model. The trion model is a highly structured mathematical realization of the Mountcastle organization principle, with the column as the basic neuronal network in mammalian cortex. The column comprises minicolumns called trions. One particular columnar network of trions has a large repertoire of spatial-temporal firing patterns, which can be excited and used in memory and higher brain functions (3). Shaw and Leng performed an experiment in which they mapped the trion model of firing patterns in that particular column onto various pitches and instruments producing recognizable styles of music. This mapping of the trions gaves insight to relate the neuronal processes involved in music and abstract spatial-temporal reasoning (3). It shows that the part of the cortex, which contains the repertoire of spatial-temporal firing patterns, can be excited by music and is utilized in higher brain functions such as spatial-temporal thinking in mathematics.
In conclusion, my research into math and music does seem to suggest that music enhances mathematics skills. Music targets one specific area of the brain to stimulate the use of spatial-temporal reasoning, which is useful in mathematical thinking. However, as to the question of whether or not music is the magical portion that will elevate anyone's ability to do math, the answer unfortunately . . .would be no. Just because most mathematicians are fond of music, dosen't mean that all musicians are fond of mathematics. I found a letter posted on the web written by a fourteen-year-old overachiever to a mathematics professor. The student expresses his fraustration that even though he is an excellent musician, math is one of his weakest subjects. In math, he is not making the grades that he needs to stay in a certain prestigious academic program (4).
This letter seems to suggest that listening to music, or being able to master a musical instrument does not automatically guarantee that one can perform well in math. In other words, there are many musicians who are good in music but not in math. Music is a lot more than notes conforming to mathematical patterns and formulas. Music is exhilarating because of the intricacies of the patterns that occurs. Whether or not these patterns resemble math has no relevance to many musicians. More often than not, musicians are inclined to practice music because of the wonders and awe that they feel for music even if they are not aware of the math that is in music.
WWW Resources
(1)Making the case of music education
(2)Music on the mind
(3)Spatial-temporal versus language-analytical reasoning: the role of music training
(4)Letter written by a young musician
Comments made prior to 2007
Are you basing your conclusion on a letter written by a fourteen year old student? If someone has low math skills there are many factors that could lead to that. There aren't any studies that I've seen stating that the study of music will raise math scores or create a situation where a student will excel in mathematics. The studies will show that the study of music will enhance the brain function and thought processes involved in mathematics and scientific thinking. A result COULD be increases functionality in mathematics. Whether a student is aware of this or not is irrelevant. A student doesn't have to be aware of learning for learning to take place ... Aron, 24 January 2007
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| This paper reflects the research and thoughts of a student at the time the paper was written for a course at Bryn Mawr College. Like other materials on Serendip, it is not intended to be "authoritative" but rather to help others further develop their own explorations. Web links were active as of the time the paper was posted but are not updated. |
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Comments
Can Kpop music help? I find
Can Kpop music help? I find that listening to kpop music can help me focus a lot on my studies like math.
music & math
Don't get me wrong, you have to be strong and confident to be successful in just about anything you do - but with music, there's a deeper emotional component to your failures and successes. If you fail a chemistry test, it's because you either didn't study enough, or just aren't that good at chemistry (the latter of which is totally understandable). But if you fail at music, it can say something about your character. It could be because you didn't practice enough - but, more terrifyingly, it could be because you aren't resilient enough. Mastering chemistry requires diligence and smarts, but mastering a piano piece requires diligence and smarts, plus creativity, plus the immense capacity to both overcome emotional hurdles, and, simultaneously, to use that emotional component to bring the music alive.
Before I started taking piano, I had always imagined the Conservatory students to have it so good - I mean, for their homework, they get to play guitar, or jam on their saxophone, or sing songs! What fun! Compared to sitting in lab for four hours studying the optical properties of minerals, or discussing Lucretian theories of democracy and politics, I would play piano any day.
But after almost three years of piano at Orpheus Academy, I understand just how naïve this is. Playing music for credit is not "easy" or "fun" or "magical" or "lucky." Mostly, it's really freakin' hard. It requires you to pick apart your piece, play every little segment over and over, dissect it, tinker with it, cry over it, feel completely lame about it, then get over yourself and start practicing again. You have to be precise and diligent, creative and robotic. And then - after all of this - you have to re-discover the emotional beauty in the piece, and use it in your performance.
I would say that your
I would say that your assertion that Chemistry/Science/Philosophy don't require creativity to be false. The application of maths that many people reach are just that, applications of someone else's thoughts. The same way playing a 4 chord progression in a pop song would be looked down upon by a classically trained pianist for a lack of creativity, a mathematician would say that the Calc 1 class one has to take in college is elementary. However, if you were to look into a field like Topology, you would be blown away by the creativity required. The pure abstract thought processes are not dissimilar for Maths and Music, the two fields funnel their creativity into separate topics, ands thats fine. To say that Maths/Chemistry/Philosophy don't require creativity is, quite frankly a naive statement.
3 chords are harder than they look
The best players often can't play something simple and have it sound good. How could a classical pianist look down on anyone? A lot of them are lost without a sheet of music in front of their face. Rock players *memorize* it. All of it. Jazz players also have strong and undeniable skill sets. A number of players can't just jam or just play.
Typo in The First Paragraph
You have a typo in the first paragraph, second sentence. You said, "...two seemingly unrelated subject...", but what you should've said is, "...two seemingly unrelated subjects...". Your welcome for the help:)
You have a typo. You said
You have a typo. You said “your welcome,” but you should have said “you’re welcome.”
Your= possessive
(That is your car.)
You’re= you are
(You are happy.)
While e're at it, I think you
While e're at it, I think you meant "You're welcome" and not "your welcome". :)
Music and maths
My daughter is a 19 years pianist, started piano a six. She won a number of international competitions for young pianists. She will be offered a place in some prestigious conservatories but she will decline the offer and prefers to apply for a Maths and Music degree in a top university in UK. I stii have to understand if it is because he is upset of playing the piano or if she think to have more fun in that college !! It's a fact that playing seriously an instrument by the age of 6 and for at least 7 years change plastically your brain but then we really do not know what these changes mean, what abilities they trigger. The other way round doing mathematics to learn music does not work. It is also a fact that true musicians and real artists do not come to terms with maths, they simply do not see any interest in math. The numeric terms and anrchitecture that underpin any musical score does not suffice to explain music (fortunately). We need a human being playing that score and make it alive in its own way while sticking to the composero terms and stylistic content of the work.
Music & Math Studio Supports Music & Math Correlation
There is an actual business that was established to demonstrate the correlation between music & math. This business was founded by a former band director and math educator in 2013. Since that time, scores of musicians and math students, many of whom are the same individual, have affirmed that either of these disciplines can be mastered so long as one is mastered. Music & Math Masters Studio (Woodway, Texas) now deals with motivating and training clients, ages 3 and up, how to play any musical instrument, solve any math problem efficiently, and sing any genre vocally in addition to providing professional audio and video recording services. As a retired musician and mathematician of more than 30 years, may I suggest that you begin your research by studying the various professionals who have made their life's goal or career based on uniting both music and math.
correlation between math and music
I am currently a middle school student and ever since I was born, I have been in love with music. When I got older, I noticed that math came easily after I learned my music. I have to do a Capstone project and I have chosen to prove that music does help students learn all subjects and that music connects the two sides of the brain and makes it easier to learn a subject. A video I watched on youtube says that certain music helps students learn certain subjects. another video, this one by TedEd, says that music connects the two sides of the brain and makes connections easier and faster. The research I have done so far states that there are variable that can cause music to not help, but mostly it does help. it also states that music helps people keep the information instead of forgetting it after a week or two. This is all from memory from yesterday so these facts are most likely true, but if they aren't it is because I must have mixed up information over night.
relation between maths and music
I study in harvard university
Everyone say about this relation but there is no such relation.
Everyone who say this are fools as these two words are of totally different category.
Children in my class as well in my school, who were good in maths or physics are not even having musical sense.
This is a total bullshit.
What? i'm sorry good sir, but
What? i'm sorry good sir, but i do not understand what you are trying to say. I can try to translate for the higher level audience, '' I am a student at Harvard University. Everybody talks about this relation, but it is untrue. Anybody who says this is a complete fool. These two words(?) are in totally different categories. Children in my class, as well as my school, who are good at math or physics don't have any musical sense. This is complete hogwallop.'' I hope this isn't actually professor david marshall, because this person misspelt the name. I hope this is easier to understand for everybody else.
What a compelling counter claim
Hello. I'm glad to hear you've made it into Harvard University. I am also surprised that you were accepted into Harvard with your atrocious grammar
1: I study in *Harvard *University
What does this have to do with the claim you are trying to make?
2: Everyone *talks about this *correlation^, but there is no such *correlation
Yes, there is. Many studies have gone and proven this.
3: *Everybody who *says this *is a *fool, as these two words are of totally different *categories
Excuse us for not being as smart as you. You are truly the superior person, as you are clearly a master of the English language
4: *Students in my class ^, as well in my school, who were adept in math or physics, but have no musical sense
I would hope there aren't any children in your college. Also, the argument is the music helps math, not math helps mucic
5: This is *total bullshit.
Ah yes, profanity. You certainly are the more mature of us here, so the profanity claims that you are always right.
Please, before you go and prove a point wrong, make sure you yourself can make a point.
And if we're going to state what education we have, I am a highschool student
Specious Claims
"I study in harvard university"
If this were true then you would have written, "I study at Harvard University'.
Have standards dropped that sharply at America's most esteemed tertiary education institution?
relation between maths and music
A very lucid rebuttal to the cited study. Harvard must be proud that you are its standard bearer. I can offer only anecdotal evidence, but I play three instruments and have an undergraduate degree in mathematics. Likewise, my daughter plays four instruments and plans to major in mathematics when she goes to university in the fall.
Hi Everyone, My name is
Hi Everyone,
My name is Titu), a student of Texas Southern University, Houston, Texas. This Summer, I have been approved for an undergraduate research on "Relationship between Music and Mathematics." As a violist from Africa, Nigeria precisely, I have never imagined that such relationship exist at all. May be is due to lack of access to information and internet.
I would to know and possibly talk with any professors or scholars who have carried out research(es) on this topic
Your research
What did you learn?
Relationship between music and mathematics
I think you have all spoken well of both music and mathematics. I think the biggest link between both is the "M." Memory! Recently, I was given a research assignment "Relationship between music and mathematics," and it has kept me thinking wild and wide. From my musical experiences in Africa, Nigeria. I have seen many music students crossing to other fields without any trouble, but there were solvable challenges which they faced. The point here is knowing what you have and making right choices in facing challenges. Willingness to pay the price of learning will help when you put to work your ability.
Truly, music enhances abstract thinking and analytical thinking but is not something that will come without paying the price of learning and studying. no skill will come by magic but only through self discipline. To be a good mathematician you must solve problems and to be a musician you must practice scores. Although, you have better chances of learning faster if you are a serious student and practical musician.
Please I need more articles and materials on connections between music and mathematics. We are all learning from one another's view.
Brazilian musician
Hi! Having read superficially comments and the article in question, what I know is there are strangely relational side effects from both Music and Maths and Science areas. I conduct choirs, arrange and play Hammond or keyboards and also deal with soldering, can inspect an electro appliance and fix simple causes, dealt with sound mixing in past and read well general manuals of electrical devices, having in the past also programmed computers reasonably. Well, I wouldn't say music causes my "easyness" in dealing with sound and electronics in general, neither these things I can do helped my music. Intuitively, definitely there's "something in the air". I'm aware of fact that little babies - under 2 years - hearing Classical Music, from Mozart's period, quickly develop all other habilities due to this audition. Good article, indeed, but really is a simple departure in those incipient related ideas. Nowadays, restudying all classical again in piano, mainly with Bach's Preludes and Fugues, made my jazz - or bossa, samba, funkish - improvisations much better!
Math plus (+) music
Greetings!
As I am reading statements, comments and opinions below, it does help me in my research. I am presently working with my research though I had an experience on the positive outcome on listening to classical music, I still need opinions and other research from you. Can you please help me to find other sites discussing this topic (Math and Music) for my references? By the way I got some screenshots of comments below which I could use as my reference. I love classical music! Thank you.
correlation between music and math
I'm pretty good at math, although not higher math. Didn't study that.
I play music strictly by ear - I don't read standard notation. I play piano, organ, elec an acoustic guitar, and bass guitar. Been playing for over 50 years. I understand the basics, such as key, minors, sevenths, etc., intervals, time, sharps and flats, and such. Enough to be barely conversant with other musicians. I'm really more of a, "I can't tell you, but I can show you," kind of guy.
Yet, my brain understands that there is a relationship between music and math. At the very least, it understands repeating patterns.
I do not understand how I know how to play the complex music I am able to play. As I'm learning a piece, my ears hear the correct intervals between notes that form both the chords and the melodies, with heavy emphasis on the function of the bass line. I am able to work with closely voiced harmonies and with widely spaced voicings, with the ability to determine when to use those spacings. I have recently (within the last 10 years) started incorporating more dissonance, much to the enrichment of the songs.
I guess the point I'm trying to make is that music and math are very much related. In my case, the basic musical ability was there at a early age, and a fair understanding of math seems to have enabled me to grow my understanding of music. That, in turn, has allowed me to expand my music vocabulary. It's something I understand, but - like music - I find difficult to explain.
Interest in music/maths (we say maths in Australia) connection
I really enjoyed your description of your musical experience and abilities. Thoughtful and interesting.
I am teaching myself ukulele and have rejected for so long formal training, it is crazy but I am needing to reconnect my body/ear/brain with the sound in a fundamental way, on my own.
I do hope that I will eventually be able to hear music in a similar way to you. And play!!
Thanks
maths
Math is very boring sunject without music
Bad math and music
I really suck at both maths and music. Although I was forced to learn an instument, I still hate all those pitches and beats or whatsoever. I just get a headache when I see all those black notes. However I kind of believe there is some relationship between maths and music, because all the piano players in my class get good grades in maths.
Thank You.
I am a musician and songwriter and This really helped widen my knowledge in mathematical aspects in music. Thanks for sharing!
luv ya
Thanks so much this helped me a lot luv u ( ; ( ;
What is the purpose if there is no connection to be made
Honestly I hate math. But, no matter where I go many people expect me to be an amazing mathematician because I play two instruments. I am not very good at math and have been playing for years so if this study is true than it is only plausible for certain groups of people. Many people have said that there does not have to be a connection for a student to begin to excel at math but that is not true in many people's cases. If a person does not understand a connection then they cannot learn from the other thing like Mozart and math. Many people agree with me ad say that there may be some things affected but not as much as researchers think. They seem to think that all humans are the same, if they truly wanted to make a completely plausible statement they would take into account age, grade, time spent doing each, and opinion towards each subject. They must have studies with percentages or fractions instead of saying it helps all people because that is not true.
Hope people agree with me.
No necessary connections between music and maths ability
My daughters both study string instruments. One of them is not good at all in maths. The other is very good at it. I personally think that there is no such a link and that playing music does not make you necessarily more intelligent nor necessarily good at maths. There is may be the rule that if you play music this is because your parents have tried to give you a better education and then you can be good at math, literacy, etc. just because you are motivated to study in general. Just think of how many musicians there are in the world. They are not all genius in other disciplines than music.
On the contrary, I think that studying music can improve your social skills when you play in bands, sing in choirs, etc. This yes.
It is also true that music has positive effect on your mood, motivates you, gives you energy. And this, more than anything else,is important in your life and can may be help you when you struggle with maths. At least I hope you for my daughter that is really struggling with maths.
only two instruments? i play
only two instruments? i play 6
irrelevant
Really? Not the point of the discussion.
sightreading/math/classical
No. Sight reading makes you better at math. The sound is not important, but rather the analysis of the notes. Because the notes are all the same, unlike the alphabet, the skill is not necessarily 'reading' but a mathematical deciphering of spaces. I can play difficult pieces by Liszt and Chopin well if I've heard them before, but I'll get disoriented sight reading a simple nocturne. I am also horribly bad at math. Even basic addition and subtraction was difficult to grasp for me. im not trying to have a pity party here its just the honest truth. i would like to believe sight reading music can challenge the mathematical function of your brain and build that skill. Much like your writing, grammar, and vocabulary influence interpretation.
Disappointing
As a pre-service music educator looking for a research topic I was sorely disappointed in this paper. Not only is it poorly written, but the sources are questionable and scarce. I truly believe that music can be used like any neuroplasticity exercise to increase aptitude and this article gives it a bad name. There are too many quack researchers out there making ridiculous correlations between just listening to music and magically having better concentration/higher math scores/faster reading levels/etc. Hopefully some readers (though based on the comments I don't know why I'm bothering saying this) will consider some more reliable evidence for using music as a means of increasing basic pattern-recognition skills, among others, before giving up on the subject.
hi
THATS COOL REALLY HELPED MY RESEARCH!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Music & Math
I found this article very interesting. I don't know much about statistics, but I do know that when I play Handl with my math learning centers and use it as a background music while they are exercising visual skills and using my multisensory approach to
learning process steps, they (adults and kids) are having fun. My scores on the GED Math test for adults I work with have gone up. Proof is in the pudding for me.
Hello! I'm new here, so just saying hi :)
I really like your forum here. So I decided to be a part of it :)
And here I am saying HELLO EVERYBODY!! :D
At Last!
Finally--an explanation for why so few musicians are good at balancing their checkbooks!
Mathe and Music
I am going to findout relation between math and music but I still coulldnot get exactly point.
can maths music help students to learn
yes it can help students
Research
I will ask my students in college to conduct study on this case. Thanks for posting. Hope to receive more materials from this website.
languages,verbal/literal,math,music
Do you wanna talk? I do! I`m at . I`ve taken algebra, geometry+trig, pre-calc, I also read a book by H. Jacobowitz called Computer Arithmetic, it contains 5 rules for formulating a system of positional notation. It really epiphany-ically enlightened me: The hindu-arabic number representation we use expressed as a polynomial, a better understanding of synthetic division, good coffee, plenty of Led Zeppelin and I`m well on my way to solving the n vrs. np conjecture, the other Millenium Prize probs, unifying the strong nuclear/electro-weak/gravitational forces in an efficient equation...well maybe the coffee was too strong. Rhythm, pitch expressed as frequency, tempo, cadence,verbal-literal and mathematical language, ST &LA reasoning, now that`s an awesome unified field theory!! I love it !!! And I love you too!! contact me if you`d like. Have a Blessed day!!
From what I can see, your
From what I can see, your writing skills are lacking.
languages, verbal/literal, math, music
I read your post and all I could understand was Led Zeppelin. And I love you. And Have a Blessed day. lol hysterical.
The Music and Math code
There is most definitely a pattern identity and a reason why there are 88 keys on the piano. Its very simple. I've seen it on the 88 logic DVD sponsored by Yamaha.
The reason for the 88 keys is based on the division of the 52 white keys and the 36 black keys into the 7 octaves show the reserve keys at the bottom of the keyboard. Adding of these reserve keys and applying them as leverage on top of the 13 keys per octave creates the 16 keys needed to identify the reason for the 88 keys as the countable ratio which shows how to play the piano.
This is further proved by the division of the 88 keys into 4 quarters and using those key placements as a direction to find the reserve keys by placing them into one octave, or the top octave. Either way, you find the ratio. Then you can understand true piano flow to finding (going with the grain) the major influence keys and the minor (going against the grain). It's that simple. For more info read my blog stephenrmackay.wordpress.com or go to www.pianodiy.com.au. It's all there in plain english, book and DVD form for those who learn kinesthetically.
Math & Music= MUSACtivities!
This is a very interesting read, seeing that we, MUSACtivities have designed our programs based on this very correlation. We appreciate many of these comments because they supplement previous ideas or sparked the development of new ones. As a young company, any information, suggestions, comments & experiences help a great deal. If anyone is interested in learning more or being an active resource, PLEASE visit .
Also, you can find us at www.facebook.com/MUSACtivities or www.twitter.com/MUSACtivities
We have a responsibility to educate our youth and the desire to do it through music!
comment....
Hi, im a graduate school student working on my research title proposal. i was inspired by your article and i have in fact proposed a research title about music integration in the study of chemistry... i noticed that my students learn better when i introduce science concepts through a song or when they work on composing a song with a melody to generalize, summarize or synthesize the things that they have learned.
however, i am still processing this idea and trying to gain more information/ resources to support my proposal.
can anybody help me? please...
thanks
music and brain
Music has been used since ancient times to calm the nerves, reduce stress etc.
Knowing now the effect of music on sequential reasoning, one wonders why there is no emphasis on music education, starting from elementary schools.
Won't that be a cost effective investment?
Now that there is a cry about the shortages of engineers and mathematicians, this seems like a smart strategy for securing the future while at the same time enhancing brain health.
Unfortunately, less and less children are exposed to music nowadays because of lack of funds.
The only exceptions are those whose parents can afford private lessons
WOW
Bravo to you. I do agree with you in all levels. Im a medical physician doing a neurology residency in Philadelphia. Music does help you increase your math skills. Reading this article took me years back when i was in high school, I used to practiced calculus and trigonometry listening to music, but not classic though, very loud high beat kind of music. IT helps me concentrate, silence doesnt do it for me, is kind of distracting actually. With music, i create my own universe; Pascal said that we enjoy music because it causes us to count without our knowing it, that is true.. Most people find it distracting, and actually it is not at all... cause Even if we aren’t paying close attention, our brain counts each note and change of rhythm. This counting occupies the part of the mind that hinders our creative abilities. Like a soothing lullaby, music puts the worrisome unfocused part of the mind to sleep so the productive side can get to work.
pascal didnt say that lol. is
pascal didnt say that lol. is was gottfried leibniz. if pascal did say that it was a rip off haha
where are the statistical data?
I stumbled upon this article, and I wanted to see if I statistically figure out how much of a difference music makes.
Were the samples randomly chosen? It would have to be matched pairs...with vs. without the music...because people already work differently. It would have had to be a Paired T-Test or Interval...What was your confidence interval? Did your conditions check out for Independence and Normality for the mean of differences between with and without by matched pairs? What was your sample size? Where are the data?
Science Fair
This article helped me for my science fair project. Thank you!