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Studies and experiments of all different kinds are carried out worldwide every day, and all of them aim to unlock new knowledge or confirm previous findings. What can the raw data collected in these studies be used for? Well, it's used to come to conclusions and make robust generalizations.
Define Concepts Conclusions and Generalizations
What exactly is a conclusion? Well...
A conclusion is a finding drawn from a set of data in a study or experiment.
For instance, if you had run a study for headache medication and found that \(90\%\) of participants who took the real pill saw a great reduction in headache pain, compared to only \(20\%\) who took the placebo pill, you might make the conclusion that the pill was effective in reducing headaches in those in the study group.
Now, is the pill effective in reducing headaches in everyone? That really depends on how the study was conducted. This type of assertion is called a generalization.
What is the Difference between a Conclusion and a Generalization?
This question can be a bit misleading. Think of fingers and thumbs. All thumbs are fingers, right? But are all fingers thumbs? Of course not! The same is true of conclusions and generalizations.
A generalization is a type of conclusion that can be applied to all or almost all members of a larger group.
For instance, if you stopped two hundred people at random around a town and every single one of them said their favorite food was pizza, you could make a generalization that, in general, everyone in the town's favorite food was pizza.
But wait, as always, with research, there is more to it than that!
Conclusion and Generalization in Research
Considering the pizza example further, where did you stop those people? Was it outside of the local pizzeria? If so, perhaps this generalization is not such a good one to make. However, if each person asked was truly chosen at random, for instance, by picking names out of a comically large hat, then the generalization could be acceptable.
In research, it is important to be able to distinguish a conclusion drawn from one group of participants from any generalizations made about a wider group. For instance, if you put out a request for people to participate in a survey about their favorite food, are the participants a random selection from the wider group?
No! The participants are just the people who decided to respond. Maybe pizza lovers in the town are just more passionate about their favorite food than hamburger lovers are, and so were more likely to respond. Always take surveys with a pinch of salt. Consider how the survey group was chosen, or if it was even 'chosen' at all!
There is a fine line then between a simple conclusion and a generalization. Think about these concepts in your own life. Maybe in the town you live in, everyone thinks hot dogs are the best food. Is it a good generalization to say that everyone in the world must love hotdogs just as much too?
When considering if a generalization can be made of a conclusion found of a group of participants, there are several things to consider.
The first is the randomness of the sample group. If you wish to make a generalization about a whole population, it is imperative that the participants are chosen at random.
The second is the size of the sample group. If you picked ten people in a town of \(3000\), your results would not likely show a distribution of data representative of the entire town.
The third is the internal validity of the study itself. In other words, whether there are any other factors that might have influenced the findings outside of the specific variable you are testing. If a study testing the effect of screen time on mental health is conducted by taking the phones from a sample group for a month, can the results conclusively show that screen time was the sole cause of an uptick in good mental health? No, because it could have been a number of things other than screen time that caused it, such as a reduction in time spent on social media.
Drawing Conclusions From Data Statistics
How exactly are conclusions drawn from statistical data? It's not usually as cut and dry as every single person asked, saying pizza is their favorite food. What if \(90\%\) of people say that? Is it a valid conclusion to say that most of the people in the town's favorite food is pizza? Probably, but what if only \(55\%\) say that? That could just be a quirk of the data. Maybe if you asked \(200\) different people, you would find something different.
Well, this is where statistical analysis comes in. Statistical analysis can be used to find the probability that a certain set of findings could occur through random chance. If \(90\%\) of people said, their favorite food was pizza? There's a very small chance that could happen by chance, but \(55\%\)? Well, that's a bit less far-fetched.
Ultimately, if a certain finding is determined to have a small enough chance of being due to random chance, the findings are considered a valid conclusion, but if it's too large, then a conclusion can't be drawn.
This can all be complicated to do, but luckily there are some methods that you will learn in this course to be able to carry out this sort of statistical analysis on different types of data sets.
Statistical Conclusions and Generalization Examples
See if you can work out from these statements whether the generalization or conclusion made is valid or invalid.
(a) Voters leaving the only polling station in a small town are asked which party they voted for. \(70\%\) said democrat, \(24\%\) said republican, \(4\%\) said they voted for another party, and \(2\%\) said they spoiled their ballot.
From the data, it was concluded that most of the town supports the \(democrats\).
Solution:
This conclusion is invalid. The only people surveyed are those who voted, those who chose not to, or could not, vote are not surveyed and therefore are not accounted for.
A more accurate conclusion would be: most of the voters in the town support the democrats.
(b) \(100\) people are selected at random from the local gym's Facebook page to take a survey about their fitness goals. \(60\) of them said that they wanted to be healthier, \(25\) of them said that they wanted bigger muscles, \(10\) said that they wanted better stamina, and \(5\) said they wanted to meet people.
From that data, it was concluded that most people joined the gym to be healthier.
Solution:
This conclusion is invalid. The people surveyed were selected at random but only from a pool of those on the Facebook page. Certain people are more likely to use Facebook, such as younger people, and therefore the selection was not truly random, and the findings could be skewed.
(c) Thirty pupils in a year group of \(100\) are chosen at random for a study. Fifteen of them are asked to delete Instagram from their phones for \(30\) days, and fifteen of them are not. They are all interviewed daily on the topic of their mental well-being. The study found that \(11\) out of the \(15\) with Instagram deleted had a significant uptick in their mental well-being, compared to \(3\) of the \(15\) without it deleted.
From the data, it was concluded that, in general, deleting Instagram from their phones would be a benefit to the mental well-being of pupils in this year group.
Solution:
This generalization is valid. The sample was chosen at random, the sample size was large relative to the size of the year group, and the control variable (the presence of Instagram on pupils' phones) was the only thing that changed, so the study was internally valid.
d) For the study undertaken in part c), would the following generalization be valid?
In general, deleting Instagram from their phones would be a benefit to the mental well-being of pupils this age across the country.
Solution:
This generalization would be invalid. There is too much variance in the situation between pupils in this year group at this particular school and pupils in other year groups at other schools across the country. The sample was chosen only from this year group at this school, so any findings are not applicable to other year groups at other schools.
Generalization and Conclusions - Key takeaways
- A conclusion is a finding drawn from a set of data in a study or experiment.
- A generalization is a specific type of conclusion that can be applied to most in the group from which the sample was taken.
- For a generalization to be made of a population from a sample group, the sample must be selected at random, must be large enough, and the study or experiment itself must be internally valid.
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Frequently Asked Questions about Generalization and Conclusions
What is a conclusion in general mathematics?
A conclusion is a finding drawn from a set of data in a study or experiment.
Are generalizations and conclusions the same?
A generalization is a specific type of conclusion that can be applied to most in the group from which the sample was taken.
What are conclusions and generalizations?
A conclusion is a finding drawn from a set of data in a study or experiment. A generalization is a specific type of conclusion that can be applied to most in a group from which the sample was taken.
What is an example of a generalization?
If you asked 300 people in your town at random what their favorite food was and most said pizza, a generalization would be to say 'The people in the town's favorite food is pizza'.
Why is generalization important?
Generalization lets us infer things about large populations without having to study the behaviour of every single constituent of the population.
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