# Problem solving and data analysis subscore on the SAT test

## Included topics: ratios rates and proportions; percentages; units and unit conversion; linear and exponential growth; table data; data collection and data inference; center spread and shape of distributions; key features of graphs; scatterplots.

Problem solving and data analysis subscore is one of the three SAT math test subscores.

**Problem solving and data analysis subscore includes the following algebra topics: **ratios rates and proportions; percentages; units and unit conversion; linear and exponential growth; table data; data collection and data inference; center spread and shape of distributions; key features of graphs; scatterplots.

**The skills tested in heart of algebra subscore:**

- Problem solving and data analysis subscore tests solving problems you encounter in the real world situations.
- The skills tested in problem solving and data analysis subscore include representing problems, applying units rates, analyzing relationships and draw conclusions about information.

**About the test:**

- Problem solving and data analysis questions comprise 29 percent of the SAT math test (17 of 58 questions).
- Problem solving and data analysis subscore questions include multiple choice questions and student produced response questions.
- The use of a calculator is permitted for all questions.
- The grade of problem solving and data analysis subscore is reported on a scale of 1 to 15.

Before learning this subscore, you need to learn the heart of algebra subscore that includes the basic topics needed for understanding this subscore.

### Ratios, rates and proportions SAT topic

This subject includes 3 connected topics that deal with representing relationships using division: **ratios, rates and proportions.**

Since ratios, rates and proportions are written as fractions, before learning this topic you should learn solving fractions topic.

**A ratio** is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented using a colon as a:b or as a fraction a/b. For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to girls in the class is 11:22, reducing 11:22 gives 1:2. The ratio of girls to boys is 22:11, reducing 22:11 gives 2:1=2.

**Equivalent ratios** are ratios that express the same relationship between numbers. Two ratios are equivalent if we can reduce or expand one ratio and get the other ratio.

**Complementary ratios** are ratios that add up to a whole that is 1.

**There are 2 types of ratios: ** a part to part ratio and a part to whole ratio.

**A rate** is a quantity compared to another related quantity, where the quantities have different units. For example: The speed per hour is the rate measured by the number of miles per a unit of time of 1 hour.

**A unit rate** is a rate compared to a single unit quantity (the denominator is 1).

**To calculate the unit rate,** divide the total of one quantity (the numerator) by the number of units of the other quantity (the denominator).

** ****A proportion** is an equality between 2 or more equivalent ratios. The proportion between a, b, c and d can be represented using a colon as a:b=c:d or as a fraction a/b=c/d. For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, we need to add 9 cups of flour: 3/1=9/3 or 3:1=9:3

**We can write proportions in 2 ways:** There are same units above the divisor line and below the division line **or **there are same units at the nominator and the denominator of each ratio.

**Solving proportions is done using the cross product method** which involves multiplying the numerator of one fraction by the denominator of another fraction and then equaling the multiplications.

### Percentages SAT topic

**A percentage** is a number or a ratio expressed as a fraction of 100 and represents a part to whole relationship. The different forms of writing percentages are a decimal number, a fraction, and a ratio. Note that a percent term is used after we write a value (for example: 10 percent).

**Complementary Percentages** add up to 100%, where 100% refers to the total.

A percentage= 100%- complementary percentages

**A percentage formula is**

percentage % = 100 * the value to be expressed

__________________________

total

(The value to be expressed can be smaller than the total or bigger that the total).

**To convert a percentage to a decimal fraction, **remove the percent sign % and divide the percentage by 100:

Percentage as a decimal fraction=percentage without the % sign/100

**The percentage of the change is equal to **the difference divided by the initial value multiplied by 100:

The percentage of the change= (The final value – the initial value ) *100

_________________________________

The initial value

**Percentage data can be given in a table:**

- All percentages must sum to 100%.
- Each percentage in the row can represent a value to be expressed. Calculate the value to be expressed of each row by multiplying its percentage as a decimal by the total of the table.

Click here to learn the required skills and solve questions about percentages on the SAT test.

### Units and unit conversion SAT topic

**A unit **is used to measure a quantity, while different kinds of quantities measured in different units.

**Unit conversion** changes a given quantity to a different unit of measurement, so that the relative amount does not change. Unit conversion is calculated with unit equivalencies.

**Unit equivalencies** show how the units of the same kind of quantity relate to each other.

Time Unit equivalencies measure time intervals that express duration (for example: 1 minute= 60 seconds).

Distance Unit equivalencies measure length (for example: 1 kilometer=1,000 meters ).

Mass Unit equivalencies measure weight of objects (for example: 1 kilogram=1,000 grams).

Volume Unit equivalencies measure capacity of an object or space in three dimensions (for example: 1 liter=1,000 milliliters).

**Unit conversion is calculated** by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

**To Convert between 2 units a and c using a third unit b,** we need to multiply by 2 conversion factors: a conversion factor between unit a and unit b and a conversion factor between unit b and unit c.

**To Convert units that appear within rates,** we need to construct one or two conversion factors to cancel the units that are given and not desired and create the units that are not given and desired.

### Linear and exponential growth SAT topic

**Linear growth **occurs then a variable is growing by the same amount in each unit of time. The linear growth formula is y=mx+b, where b is the initial value and m is the constant rate of change.

**Exponential growth **occurs when the variable is growing by the same __relative__ amount in each unit of time. The exponential growth formula is y=a(b)^{x}, where a is the initial value, b is the number that is multiplied and x is the number of the multiplications.

**The difference between linear and exponential growth:** Linear growth is always at the same rate, while the rate of the exponential growth increases over time.

**Model a table with a linear equation** if the value in the table changes by a constant amount.

**Model a table with an exponential equation** if the value in the table changes by a constant multiplication factor (the common factor).

**Modeling a scenario with linear and exponential functions** questions require determining the type of the connection (linear or exponential) and the direction of the connection (increasing or decreasing).

- If the change is at a constant rate (by a constant), the relationship is linear.
- If the change is by a percent from the initial value, the relationship is linear.
- If the change is by a factor, the relationship is exponential.
- If the change is by a percent (from the previous value), the relationship is exponential.

### Table data SAT topic

**A frequency table** is a table that shows the number of times the items occur.

**A two- way frequency table** displays frequencies for two variables so that one variable is represented by rows and the other variable is represented by columns.

**To calculate a ratio** from a two- way frequency table we need to find the 2 relevant values in the table and divide them. If possible, simplify the result.

**To calculate a percentage** from a two- way frequency table we need to find the 2 relevant values in the table and plug them into the percentage formula.

**To calculate a probability** from a two- way frequency table we need to find the 2 relevant values in the table and divide them.

**We can find missing values in tables:**

Finding the total of a table.

Finding missing values inside of a table using data from other fields of the table

Finding missing values inside the table using a given ratio or probability value.

Click here to learn the required skills and solve questions about table data on the SAT test.

### Data collection and data inference SAT topic

**Data collection**

**Data collection** is a process of collecting and measuring information on variables of interest, that enables the researcher to test hypotheses and evaluate outcomes.

**Data can be collected with a sample or with a controlled experiment:**

**A sample** is a small group that is selected from a large population by using a pre- defined sampling method. The sample must be representative and random.

**A controlled experiment** is an experiment made on an experimental group, while one factor that is being tested is changed by the researchers and all other factors are held constant.

Each controlled experiment must have a **control group.** In the control group we don’t change the factor that is being tested in the experimental group. The participants of the control group must be randomly selected and must closely resemble the participants in the experimental group.

**Data inference**

**Data inference** is a generalization about a population that is based on statistics calculated from a small group (a sample) that is drawn from that population.

**An estimate** is a process of finding a value of a population that is close enough to the right value, by performing a sample on a part of that population.**A sample proportion** is a variable that is calculated from the sample, that we assume reflects the whole population.**The estimate formula: **estimate= sample proportion * population

**A margin of error** is the degree of error in results received from random sampling surveys, it exists since the sample does not exactly match the population.**The range formula:** range= estimate ± margin of error

### Center, spread and shape of distributions SAT topic

**Center, spread and shape of distributions** are statistical measures that describe data sets, they are called summary statistics.

**A center** of a data set is a way of describing a location. We can measure a center of a data in 3 different ways: the mean (average), the median and the mode.

**A spread **of a data set describes how similar or varied the set of the observed values. We can measure a center of a data in 2 different ways: a range and a standard deviation.

**Center measures**

**Mean** is an average value of a given data set.**Mean = average = sum of the values / number of the values**

**The median** is the middle number in a sorted in ascending order data set (the median is the value that splits the data set into two halves). To calculate the median, arrange the values in an ascending order, count them and calculate the median. If the number of values is odd, the median is the middle value. If the number of values is odd, the median is the average of the two middle values.

**The mode** of a data set is the number that occurs most frequently in the set. To determine the mode, order the numbers from least to greatest, count how many times each number occurs and determine the mode. If no value appears more than once in the data set, the data set has no mode. If a there are two values that appear in the data set an equal number of times, they both will be modes etc.

**Spread measures**

**The range** measures the spread of a data inside the limits of a data set, it is calculated as a difference between the highest and lowest values in the data set. The larger the range, the greater the spread of the data.**range= the highest value – the lowest value.**

**The standard deviation** is the measure of the overall spread (variability) of a data set values from the mean. The spread is measured as the distances (absolute values) from the mean of each value of the data sat. The more spread out a data set is, the greater are the distances from the mean and the standard deviation.

**Outliers**

**An outlier** is a value that is very different from the other values, so that it lies an abnormal distance from other values and is far from the middle of the data set.

- Mean: Removing a big outlier will reduce the mean value and removing a small outlier will enlarge the mean value.
- Median: Removing a small outlier will enlarge the median; removing a large outlier will reduce the median (the median will be the same if the values in the positions after the removal are equal to the values in the positions before the removal).
- Range: The value that will replace the outlier will be less distant therefore the range after removing the outlier will be smaller.
- Standard deviation: Since the outlier is a value that is far from other values and the mean, its removal will reduce the spread of the data and the standard deviation.

### Key features of graphs SAT topic

**A graph is defined as** a pictorial representation of data or numeric values in an organized manner. Using graph enables us to represent large amounts of data in visual form for easy understanding.

**The 2 types of questions about graphs** are interpreting given graphs or selecting a graph based on a verbal description.

**The common graph types** on SAT are bar graphs, dot plots, histograms, line graphs and scatterplots. Scatterplot subject is covered on scatterplots page.

**A Bar Graph** is a graphical display of data using rectangular bars (columns) of different heights, so that the height of each bar determines its value. The structure of the bar graph: the y axis contains values, and the x axis contains categories or time periods.

**A dot plot** is a simple type of graph that shows the frequency with which items appears in a data set. It displays data items as dots above values (or categories) on the x axis (each data item is represented with a dot above its value or category).

**A histogram** is a frequency bar graph where the data is grouped into ranges. The x axis presents the data ranges and the y axis presents the frequency (the number of values that fall into the specific range.

**A line graph** includes a line that connects individual data points together. Line graphs are used to show changes over periods of time, so that the x axis represents time values (like years).

### Scatterplots SAT topic

**A scatterplot** is a graphic representation of a data set of observations (each observation includes x and y variables and is represented by a dot in the xy plane). The purpose of the scatterplot is to visualize the relationship between the variables x and y to determine if there are patterns or correlations between the two variables.

**Scatterplots can visualize the following features of the relationships:**

The type of the correlation between the variables– positive or negative correlation or no correlation.

The type of the data pattern- linear (straight) or nonlinear (curved).

The strength of the relationship between the variables– strong or weak correlation.

Unusual features in the data.

**The line of best fit** can be drawn through the area of the dots, the line of best fit represents the trend of the relationship between the two variables.

A line of best fit may be a straight (linear) line or a curved (parabolic) line.

If the line of best fit is straight (linear), it has a slope that represents the rate of the change and an intercept that represents the initial value.

The line of best fit can be used to predict y values that are not included in the data set. The prediction can be done in 2 ways: looking at the graph of the line of best fit or calculating the values from the equation of the line of best fit.

Calculating differencies between the given value and the predicted value: For any given x value, we can calculate the difference between its y value given in the data set (represented by the dot on the scatterplot) and the y value predicted by the line of best fit.

To calculate estimated change in y values, multiply the slope value of the line of best fit equation by the given amount of the change.

Click here to learn the required skills and solve questions about scatterplots on the SAT test.