Problem solving and data analysis Flashcards
What are ratios,proportions and rates?
Give an example for each term
1) A ratio is a comparison of two quantities. The ratio of [a] to [b] can be expressed as [a:b] or a/b
ex: Say Pepper has [3] hats and [2] scarves:
The ratio of hats to scarves is [3] to [2], [3:2], or 3/2
2) A proportion is an equality of two ratios. We write proportions to help us find equivalent ratios and solve for unknown quantities..
ex: we need 3 cups of water per 1 cup of flour. How much flour do we need for 6 cups of water?
3) A rate is the quotient of a ratio where the quantities have different units.
ex: Say a swallow flies [45] meters in [3] seconds. If we wanted to find the rate at which the swallow is flying, we could set up the ratio of meters to seconds. If we find the quotient, we’ll have our rate: 15 m/s
What are the 2 common types of ratios?
Give an example
part-to-part and part-to-whole ratios
ex:
For example, if we’re making lemonade:
The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amounts of two ingredients.
The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.
Say there are three ingredients in the lemonade: lemon juice, sugar, and water. We’re given the following information:
For every [1] cup of lemon juice, we need [1] cup of sugar.
For every [1] cup of lemon juice, the recipe will yield [6] cups of lemonade.
We have enough information to write two ratios:
[1] cup lemon juice : [1] cup sugar (part-to-part)
[1] cup lemon juice : [6] cups lemonade (part-to-whole)
But what if we wanted to find the ratio of lemon juice to water?
Because the ratio of lemon juice to sugar is [1:1], we know that sugar will have the same part-to-whole ratio as lemon juice. So, the ratio of sugar to lemonade is also [1:6].
Since sugar and lemon juice both have a part-to-whole ratio of [1:6], the ratio of sugar and lemon juice to the total lemonade will be [2:6].
This means that the ratio of water (the only other ingredient!) to lemonade must be [4:6] in order to complete the whole.
So, we can identify the ratio of lemon juice to water as [1:4].
What are rates used for? And what are the common rates?
Rates are used to describe how quantities change. Common rates include speed (distance/time) and unit price (total price/units purchased)
We can then use that rate to predict other quantities, like how far a train, traveling at the same rate, would travel in [5] hours
What is unit conversion?
Unit conversion lets us change the units in which a measurement is given.
What are percentages?
A percentage is a ratio out of [100] that represents a part-to-whole relationship. Percent ([\%]) means parts per hundred.
How to calculate a percent value?
% = (part/whole)*100
How to translate percentage word problems into arithmetic?
what does these key words represent?:
“what”
“is”
“of”
“percent”
“what” means x
“is” means =
“of” means multiplied by
“percent” means divided by 100
36 is what percent of 60 –> 36=(x/100)*60
how to calculate a percent change?
Find the difference between the initial and final values.
Divide the difference by the initial value.
Convert the decimal to a percentage by multiplying the quotient by [100].
What’s a shortcut for converting percentages to decimals?
Remove the [\%] symbol and move the decimal point left [2] places.
The sum of all parts of a whole is what [in terms of percentage]?
[100\%].
What are center, spread, and shape of distributions?
Center, spread, and shape of distributions are also known as summary statistics (or statistics for short). These measurements are used to concisely describe data sets.
Center describes a typical value of in a data set. The SAT covers three measures of center: mean, median, and occasionally mode.
Spread describes the variation of the data. Two measures of spread are range and standard deviation.
How do I find the mean, median, and mode?
And what each measures of center represent?
1) The mean is the average value of a data set. sum of values/number of values.
2) The median is the middle value when the data are ordered from least to greatest.
If the number of values is odd, the median is the middle value.
If the number of values is even, the median is the average of the two middle values.
2) The mode is the value that appears most frequently in a data set. A data set can have no mode if no value appears more than any other; a data set can also have more than one mode.
How do I find the range and standard deviation?
And what each measures of center represent?
1) The range measures the total spread of the data; it is the difference between the maximum and minimum values.
range= maximum value-minimum value
A larger range indicates a greater spread in the data.
2)Standard deviation measures the typical spread from the mean; it is the average distance between the mean and a value in the data set.
Larger standard deviations indicate greater spread in the data. So the greater the range, the greater the standard deviation
What is an outlier? why outliers are often removed from data sets?
An outlier is a value in a data set that significantly differs from other values. The inclusion of outliers in data sets can greatly skew the summary statistics, which is why outliers are often removed from data sets.
What is the effect of outliers on the range and standard deviation?
The inclusion of outliers increases the spread of data, leading to larger range and standard deviation. Conversely, removing outliers decreases the spread of data, leading to smaller range and standard deviation.
What is the effect of outliers on the mean?
The removal of an outlier is guaranteed to change the mean.
If a very large outlier is removed, the mean of the remaining values will decrease.
If a very small outlier is removed, the mean of the remaining values will increase.
What is the effect of outliers on the median?
an outlier does not affect the median of a data set as strongly as it affects the mean. As such, the removal of an outlier can still change the median, but that change is not guaranteed.
If a very large outlier is removed, the median of the remaining value will either decrease or remain the same.
If a very small outlier is removed, the median of the remaining value will either increase or remain the same.
What are data representations problems?
Data representations problems ask us to interpret data representations or create data representations based on given information.
What are bar graphs, dot plots, and histograms?
1) Bar graphs
On a bar graph, the sizes of the bars are related to the size of the quantities: the larger a quantity is, the taller or longer the bar representing it is.
2) Dot plots
Dot plots use dots to represent the frequency with which particular values occur. Dot plots are usually used for low, easily countable frequencies because it’s impractical to draw or count many dots.
3)Histograms
Histograms use bars to represent the frequency at which a range of values occurs. Histograms are useful because it’s often impractical to list every possible value independently.
-Link to look how each data representation looks like(go down): https://www.khanacademy.org/test-prep/v2-sat-math/x0fcc98a58ba3bea7:problem-solving-and-data-analysis-easier/x0fcc98a58ba3bea7:data-representations-easier/a/v2-sat-lesson-data-representations
What are line graphs?
What are some key phrases to look out for?
Line graphs usually show how quantities change over time.
Keys to look out for(go down)+ look at the example:
https://www.khanacademy.org/test-prep/v2-sat-math/x0fcc98a58ba3bea7:problem-solving-and-data-analysis-easier/x0fcc98a58ba3bea7:data-representations-easier/a/v2-sat-lesson-data-representations
What are scatterplots? What do the variables usually represent?
A scatterplot displays data about two variables as a set of points in the [xy]-plane. Each axis of the plane usually represents a variable in a real-world scenario.
What is the line of best fit? (scatterplot)
The line of best fit describes the general trend based on all of the points.
What are linear and exponential growth problems? Just give an example
Alphonse’s number of followers increases by [10] each month.
Bekah’s number of followers increases by [10\%] each month.
Linear and exponential growth problems are all about understanding and comparing scenarios like the ones above.
How can we determine if a relationship is exponential or linear given a table?
When we’re given a table of [(x,y)] values, for a given change in [x]:
If the change in [y] can be represented by repeatedly adding the same value, then the relationship is best modeled by a linear equation.
If the change in [y] can be represented by repeatedly multiplying by the same value, then the relationship is best modeled by an exponential equation.
What are some common phrases in linear and exponential growth problems and to look out for?
Go to this link: https://www.khanacademy.org/test-prep/v2-sat-math/x0fcc98a58ba3bea7:problem-solving-and-data-analysis-easier/x0fcc98a58ba3bea7:linear-and-exponential-growth-easier/a/v2-sat-lesson-linear-and-exponential-growth
Did you get all correct?
What are probability and relative frequency problems?
And what does the two-way frequency tables include also?
The SAT will ask you to calculate probabilities and relative frequencies using data from two-way frequency tables.
Two-way frequency tables include two qualitative variables, one represented by rows and the other represented by columns.
What are data inferences questions?
when we have questions about a large population, we often answer those questions by surveying a representative sample: a smaller set of people whose answers can give us a good idea of how the population would answer the same questions.
How to estimate using sample proportions?
A random sample drawn from a population is representative of the population. With a representative sample, we can multiply the sample proportion by the population to get an estimate.
estimate=sample proportion*population
What is margin of error?
Its most commonly given in which form?
While we can make reasonable estimates using sample proportions, we can never be [100\%] certain that the population proportion matches the sample proportion exactly. Margins of error let us address the uncertainty inherent to sampling.
The margin of error is most commonly given as a percentage
How to calculate the range with an estimate and a margin of error?
range= estimate±margin of error
How is the margin error related to the size of a sample?
The larger a sample size is, the smaller the margin of error will be.
What are evaluating statistical claims problems?
While research results can give us powerful insights, we must carefully consider how the research is conducted, which in turn affects what conclusions can be drawn.
For example:
If a survey was given to individuals of one ethnicity, then the results of the survey are not representative of individuals of other ethnicities.
If a medical treatment is effective when tested on mice, we cannot conclude that the treatment is just as effective on humans without additional testing.
What is a sample? It provides a general information about what or whom? What is a good sample?
What is a bad sampling?
1- A subset of the group being studied
2- A sample provides information about a population without having to survey the entire group.
3-To make valid conclusions about a population, we need a sample that recreates the characteristics of the entire population on a smaller scale.
A good sample is representative and random.
Representative means that the sample includes only members of the population being studied.
Random means that every member of the population being studied has an equal chance to be selected for the sample.
4-Gather data from outside the population being studied
Gather data that overrepresent or underrepresent a subgroup of the population (not random)
What is correlation and causation?
Correlation means there is a relationship or pattern between the values of two variables.
Causation means that one event causes another event to occur.
What is a placebo?
A substance with no therapeutic effect, e.g., sugar pills
In the SAT, what is needed to establish a causal relationship
A control group. A subset of study participants or subjects that do not receive the treatment; it establishes the standard to which comparisons are made in an experiment.
