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71 Cards in this Set

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What is a ratio?
an (ordered) pair of numbers used to show a comparison between like and unlike qualities.

Written in the form "x to y", "x ÷ y" or "x:y" where y≠0
What are the parts of a ratio?
The numbers x and y are the terms of the ratio
How do we determine if two ratios are equivalent?
Two ratios are equivalent if their respective fractions are equivalent
What is a proportion?
For any two ratios a:b and c:d, a:b=c:d is a proportion

a:b=c:d if and only if ad=bc
What is constant proportionality?
If variables x and y are related by the equation y=kx then y is proportional to x and k is the constant of proportionality

y=kx is really saying that y:x=k
What are the three main types of problems dealing with percentages?
a)finding a percent of another number

b)finding a number when a percent of it is known

c) finding the percent that one number is of another
How do we solve a problem asking to find the percent of a given number?
Multiply the number by the percent as a decimal.
How do we solve a problem asking to find the number when a percent of it is known?
Divide the value of the given percent of the number by the percent as a decimal.

Ex. 240 is 40% of a number. Divide 240 by .40 to get 600 as the number.
How do we solve a problem asking to find the percent that one number is of another?
Divide the number that is the percentage of the other number by the other number and multiply by 100.
What is the interest rate?
The rate of return, expressed as a percent.
What is the principle?
The amount of money borrowed or invested
What is compound interest?
An investor earns money each year on the principle as well as on any interest previously earned.
How is interest earned/paid calculated?
By applying a compound interest to the principle.
What is the formula for finding the final value of a principle with a given rate of compound interest?
a=p(1+i/n)^nt

where p is the principle
i is the rate of annual interest as a decimal
n is the number of times interest is compounded per year
t is the number of years for the investment
What is statistics?
The science of collecting, classifying, and using data to interpret the significance ;of numerical information.
What three things should data displays always have?
1) a title explaining what the data represents

2) labels for the units or categories used in the display

3) scales that show the units and quanities of items displayed
What are the features of a Frequency Table?
various categories are listed in the table

tally marks are placed in each category to indicate the frequency with which observed data items fall into a given category

the relative frequence provieds the precent of the total number of data points represented by the frequency on each particular row of the table
What are the features of a Dot or Line plot?
easy to organize and view a set of data quickly

generally used with data sets of 40 numbers or less

scale goes horizontally. corresponding dots (or x's) are placed vertically upward of the scale
What is the range of data?
The difference between the smalles and the largest values in the data set.
What are the features of a Stem and Leaf Plot?
shows the actual data

is written in a vertical form, not horizontal

can have a dual leaf plotting (on either side of the stem) to compare two data sets
How do we make a Stem and Leaf Plot?
the tens/hundreds/thousands, etc. digits are the stem. the other digits are the leaves

if the leaves are more than one digit, separate with a comma

leaves are usually in ascending order

stem numbers are either ascending or descending
What is a pictograph?
A small icon or figure is used to represent the data values

each icon represents the same value
What is a histogram?
it has a horizontal and vertical axis. The horizontal has the values, from smallest to largest.
The vertical is the frequency

Solid rectangles are used to represent the frequencies
What can we tell about a value from the height of it's corresponding rectangle, in a histogram?
The height above an interval represents the frequency of the interval.
What is a line graph?
Line graphs are often used to show changes over a period of time. Dots are placed above the value on the horizontal axis at the frequency of that value (along the vertical axis), and the values are connected.
How can we make a line graph from a histogram?
Place a dot on the midpoint of the tops of the rectangles and connect them with line segments.
What is a Bar Graph?
a bar graph is used when the categories represent discrete groups, or at least non-adjacant intervals.

A line graph is similar to a histogram, but the rectangles are not touching.
How is a bar graph more versatile than a histogram?
We can have multiple bars for each category, to allow us to compare values for two different groups - ie. the heights of men vs women for different age groups.
What is a pie chart?
A pie chart is a circle, representing the whole, that is divided into pie-shaped sectors that represent the parts in proportion to the whole.
How do we construct a pie chart?
1. find the percentages that will be represented by each sector
2. multiply each percent by 360º to find the corresponding angle for each sector
3. draw a circle
4. draw a radius
5. draw a line segment at the corresponding angle for your first section
6. draw line segments at corresponding angles for each section
7. label the section, and make a title.
What is a measure of central tendency?
It is a numerical value used to describe the overal clustering of data
What are the three measures of central tendency?
Mean
Median
Mode
What is mean?
Mean is the average. It is the sum of the values in a set of data divided by the number of values in the set.

Denoted x bar
What is median?
The median of a set of data with an odd number of values is the middle value when the values are listed in increasing order.

The median of a set of data with an even number of valuse is the average of the two middle numbers when the values are listed in increasing order.

Denoted x hat
What is mode?
It is the value that occurs most often in a set of data
What is range?
The difference between the largest and smallest values in a set of data.
What is the upper quartile?
The median of the values greater than the median of the data
What is the lower quartile?
The median of the values less than the median of the data
What is the interquartile range?
The IQR is the difference between the upper and lower quartiles
How do we find any outliers?
Once the data is divided into quartiles and we know the IQR, we subtract 1.5 x IQR from the lower quartile, and add 1.5 x IQR to the upper quartile

any values that are beyond these two numbers are outliers
How much data is in each quarter?
Approximately 25%
How much data is between the upper and lower quartiles?
Approximately 50%
What is a box and whisker plot?
A graphical way to show the median, quartiles, and IQR of a set of data
How do we make a box and whisker plot?
After finding the median, and upper and lower quartiles, we make a vertical line for each, either above or below our horizontal axis.

Connect the lines with a horizontal line on the top and bottom (creating a box) This represents the IQR

Next, make a line above the lowest value, and one aboce the largest value. Draw a horizontal line from the box to these lines, creating the whiskers.
What is a measure of variability?
A number that describes the spread of dispertion of the data
How do we calculate standard deviation?
1. Fine the mean
2. Find the difference between each value and the mean
3. Square the differences
4. Determine the mean of the squared differences
5. Find the square root of this mean
What can we say about the standard deviation, just by looking at a set of data?
The more spread out the data, the greater the the standard deviation.

The less spread out the data, the smaller the standard deviation.
How much of the data will lie within 2 standard deviations of the mean?
At least 75%
How many standard devations of the mean are needed until almost no values lie outside?
3
What is a sample?
A collection of people or objects chosen to represent a larger collection of people or objects (the population)
What is a random sample?
A sample that is obtained such that every element in the population has the same chance of being selected for the sample.
How can we generate a random sample?
1. drawing from a hat

2. use a random number generator

3. use a random number table
How do we denote the mean of a population?
μ (mu)

we use a capital N for the total number of values in the population, instead of a small n
How do we denote the standard deviation of a population?
σ (lower case sigma)
Why do we use a random sample?
Sometimes a population is too large to find a mean and standard deviation of, so we find the mean and standard deviation of the random sample instead.
What can we say about the mean and standard deviation of a large sample?
The mean and standard deviation will be closer to the true value.
How does a histogram approach a smooth curve?
When the numbers of values in a set of data increase, the width of the intervals for the grouped data on the histogram becomes smaller
what is a normal curve?
a smooth bell-shaped curve
What is normal distribution?
The distribution of data in a normal curve.
Are all normal curves the same shape?
No, they vary according to to the mean and standard deviation.
What can we say about the curve when we have a smaller standard deviation?
The curve will be taller and thinner and the data will be less spread out.
What can we say about the curve when we have a larger standard deviation?
The curve will be lower and flatter and the data will be more spread out.
What is the 68-95-99.7 rule?
Given that the data is in a normal distribution:
within one standard deviation of the mean, we will find 68% of the data
within two standard deviations of the mean, we will find 95% of the data
within three standard deviations of the mean, we will find 99.7% of the data
How can we describe relative percomance on a test?
To give the percentage of people who did not score as high as a certain value.
What is the pth percentile of a set of a data?
A number that is greater than p% of the data and less than (100-p)% of the data
What is the 50th percentile?
The median of the data
What is the 25th percentile?
The lower quartile of the data
What is the 75th percentile?
The upper quartile of the data
How do we find the p-th percentile of n values?
1. arrange the values in increasing order
2. write p as a a decimal and calculate pn
3. if pn is not an integer, use the next highest integer for the pth percentile
4. if pn is an integer, use the mean of the values in position pn and pn+1 as the percentile
What is a z-score?
The z-score of a value x is the number of standard deviations that x is from the mean, in a normal distribution.
How do we calculate a z-score?
Z = (x-the mean)/s