Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
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Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
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Working on it
Confident
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Descriptive Statistics
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
A population is the entire group of individuals or items you want to study. It can be large (e.g., all IB students worldwide) or small (e.g., a class of 20 students), depending on the research question.
There are two types of data to be familiar with:
Categorical Variables
Non-numerical categories or labels (e.g., eye color, species, etc).
Quantitative Variables
Numerical values that can be measured or counted.
Discrete: can only take certain fixed values (e.g., number of students, shoe size).
Continuous: can take any value in a range (e.g., height, temperature).
Sampling error occurs when there is a difference between a population parameter (e.g., the average IB grade) and the sample statistic (e.g., the average IB grade at one school) used to estimate it.
This error is random and arises simply because a sample is not the entire population, and will occur even with well-designed sampling methods.
Measurement error is the inaccuracy in the data collection process. This could result from faulty instruments, poorly worded questions, or misunderstanding by participants.
For example, measuring the length of a table by counting how many hand-lengths you can fit on it is likely to yield a pretty rough estimate.
Coverage error occurs when some members of the population are not included in the sampling frame or are underrepresented, leading to a biased sample.
For example, a sample of average IB grades at Swiss schools is unlikely to represent all IB schools worldwide.
Non-response error happens when selected respondents do not participate or cannot be contacted, possibly creating bias if non-respondents differ systematically from respondents.
For example, if you send out an internet survey to estimate the percentage of the world population has internet access, then anyone who would respond "no" will not be able to access your survey.
Similarly, if you ask students to share their grade on the latest math test, those who are unhappy with their scores are less likely to respond, biasing your sample.
Random sampling means every member of the population has an equal probability of being chosen. This method reduces selection bias.
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Convenience sampling uses subjects who are easiest to reach. It is quick and low-cost but can be highly biased if the sample is not representative.
For example, sampling the heights of trees on the outskirts of a dense jungle.
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Systematic sampling involves selecting members at regular intervals from a list or sequence. For example, sampling every 5th student from an alphabetically sorted list of names.
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Stratified sampling splits the population into subgroups (strata) based on characteristics (e.g., age, gender). A random sample is then taken from each stratum, often in proportion to its size in the population.
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Quota sampling is very similar to stratified sampling, except the sample taken from each subgroup is not random.
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The mean of a numerical dataset is the average of all the values:
The mean is also sometimes denoted μ.
For example, the mean of [1,3,4,7,8] is
The mode of a dataset is the most common value in a dataset.
For example, the mode of [1,1,2,3,4,4,4,5] is 4, as it appears the most times (3 times).
If all the values have the same frequency (eg [1,2,3]), there is no mode.
The median of a dataset is the middle value when the values are sorted.
For example, the median of [5,1,7,3,10] is 5 as when sorted the list is [1,3,5,7,10].
If a dataset has an even number of values, the median is the average of the middle two. For example, the median of [2,3,4,10] is 23+4=3.5.
Mathematically, the median is the
value. Notice that if n is even then 2n+1 is halfway between two consecutive integers, indicating we need to average their values.
If we have a dataset with mean xˉ and standard deviation σ, then if we
add a constant +b to the dataset, the mean increases by b and the standard deviation does not change
scale the values by a (eg. a=2 to double all the values), then both the mean and the standard deviation are scaled by a.
Example
A dataset has a mean of 5 and a variance of 4. Each item is doubled, and then incremented by 3. Find the new mean and variance.
An original variance of 4 is a standard deviation of √4=2.
First we double the items: the mean becomes 10 and the standard deviation becomes 4.
Then we add 3: the mean becomes 13 and the standard deviation does not change.
Hence the new mean is 13 and the new variance is 41=16.
The variance σ2 of a dataset measures the spread of data around the mean.
The standard deviation σ is the square root of the variance. The advantage of the standard deviation is that is has the same units as the original data.
When you use a calculator to find standard deviation, you will see two values: Sx and σx. We use Sx when the data is a sample of a large population, and σx when the data represents the entire population.
The difference is due to the fact that a sample will usually have a smaller variance than the population, because there are fewer elements. The value Sx is larger than σx and attempts to correct for this difference based on the number of items in the sample.
Example 1
Find the variance of [1,5,6,7,11,−1].
Using technology we find Sx=4.31 and σx=3.93. Since this is the whole dataset and not a sample, we use the (smaller) value σx=3.93. The variance is then the square of this so σ2=15.4.
Example 2
A scientist samples the length of fish in an aquarium. He measures the length of 5 fish, and finds [21cm,19cm,7cm]. Estimate the variance in the length of fish in the aquarium.
Using technology we find Sx=7.57 and σx=6.18. Since we have a only a sample of the population, we use Sx=7.57⇒s2=57.3.
If we have a dataset with mean xˉ and standard deviation σ, then if we
add a constant +b to the dataset, the mean increases by b and the standard deviation does not change
scale the values by a (eg. a=2 to double all the values), then both the mean and the standard deviation are scaled by a.
Example
A dataset has a mean of 5 and a variance of 4. Each item is doubled, and then incremented by 3. Find the new mean and variance.
An original variance of 4 is a standard deviation of √4=2.
First we double the items: the mean becomes 10 and the standard deviation becomes 4.
Then we add 3: the mean becomes 13 and the standard deviation does not change.
Hence the new mean is 13 and the new variance is 41=16.
Quartiles are conceptually similar to the median, except that their are three of them: Q1,Q2 and Q3, dividing the sorted dataset into 4 equal-size parts.
Q2 is the median, dividing the datapoints in two.
Q1 is halfway between the first value and the median, at position
Q3 is halfway between the median and the last value, at position
Note: you will not need to find quartiles by hand on IB exams. These examples are for conceptual understanding only.
Example
Find the quartiles of the dataset [4,2,2,6,3,7,8,9].
First we sort the data: [2,2,3,4,6,7,8,9].
There are n=8 items, so
Q1 is at 48+1=2.25 (average of second and third).
Q2 (the median) is at 28+1=4.5
Q3 is at 43(8+1)=6.75 (average of sixth and seventh)
So the quartiles are:
ie Q1=2.5, Q2=5 and Q3=7.5.
More examples:
The interquartile range, denoted IQR, is the difference between the third and first quartile:
A value x in a dataset is said to be an outlier if x<Q1−1.5×IQR or x>Q1+1.5×IQR.
For example, in the dataset [0,7,8,9,10,11,18], the quartiles are Q1=7, Q2=9, Q3=11.
Hence
Then since 0<Q1−1.5×IQR=1, 0 is an outlier.
Similarly, since 18>Q3+1.5×IQR=17, so 18 is an outlier.
A box-and-whisker plot visually summarizes data by splitting it into quarters. The box shows the middle 50% of your data (from Q1 to Q3), and the line inside marks the median. The whiskers extend to show the spread of data, excluding outliers, which are marked with a cross.
- Minimum: smallest value (left whisker end)
- Lower Quartile (Q1): median of lower half (25% mark)
- Median (Q2): middle value of data set
- Upper Quartile (Q3): median of upper half (75% mark)
- Maximum: largest value (right whisker end)
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Datasets can be represented in frequency tables, with a row containing the values that exist in the data and a row containing the number of times each value appears.
The mean of frequency data can be calculated using the formula:
Note that n=i=1∑kfi is just the total number of points.
Example
Find xˉ without a calculator.
Using a calculator, find
The standard deviation
σx=1.33⇒ variance is 1.77.
The quartiles
Q1=2, Q2=4, Q3=4.
When data is continuous, we cannot have a column per possible value, as there are infinitely many.
Instead, we use a grouped frequency table to break up the data into specific intervals.
If all the intervals have equal size, then the modal class is the interval in which the most values fall.
We can also estimate the mean from grouped data as if it were a discrete frequency table using the mid-interval values, that is the average of the upper and lower bounds of each interval.
Example
Using the dataset [3.1,5.4,5.6,5.9,6.0,6.9], we can fill in the following table:
so the modal class is 5≤x<6.
If we only had this table (and not the actual values), we could estimate the mean using the mid-interval values:
Then
Grouped frequency tables can also be turned into histograms (aka bar graph) by drawing rectangles with base corresponding to the intervals, and heights corresponding to the frequency.
Example
The histogram
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is equivalent to the grouped frequency table
We can tell from the histogram that the modal class is 4≤x<5.
Cumulative frequency graphs are a powerful visual representation of continuous data.
The value of y at each point x on the curve represents the number of data points less than x.
We start with a grouped frequency table, and add a row for cumulative frequency, which is the number of items in an interval and all previous (lower) intervals:
To plot the diagram, we make a point from each column. The x coordinates are the upper bound of each interval, and the y-coordinates are the cumulative frequency. We then draw a smooth curve through all the points, starting on the x-axis at the lower bound of the first interval, and stopping at the last point.
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Cumulative frequency diagrams can be used to find medians, quartiles, and percentiles.
In the same way that Q1 is the value greater than a quarter (25%) of data values, the kth percentile is the value greater than k% of the data values.
Example
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Since the y-values represent the number of data points with value less than x, and 50% of values are less than the median, it is is the x-value where the line crosses 0.5× the maximum y value on the curve.
Similarly, the quartiles and percentiles are the x-values where the curve crosses:
Q1: 0.25× the max
Q3: 0.75× the max
kth percentile: 100k× the max.
Linear regression is a statistical method used to model the relationship between two variables when data is given as pairs of points (x,y). We fit a straight line (called the regression line) that minimizes the average vertical distance from the points:
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The general equation of the regression line is:
where a is the slope and b is the y-intercept.
The values of a and b can be found using a calculator.
Example
Using a calculator, we find a=3.1 and b=−0.7. So the regression line is y=3.1x−0.7.
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Pearson's product-moment correlation coefficient, denoted by r, measures the strength and direction of a linear relationship between two numerical variables x and y. Its value always lies between −1 and +1:
r=+1: perfect positive linear relationship
r=−1: perfect negative linear relationship
r=0: no linear relationship
A positive value means y generally increases as x increases; a negative value means y generally decreases as x increases. The closer r is to ±1, the stronger the linear relationship.
Pearson’s coefficient is calculated by:
Note: you do not need to use or know this formula.
On exams, r is determined using technology.
Example 1
Find r for the following data, and interpret its significance.
Using a calculator, we find r=−0.973. This indicates a very strong, negative relationship.
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Example 2
Find r for the following data, and interpret its significance.
Using a calculator, we find r=0.11. This indicates essentially a weak positive relationship, but there is almost no correlation.
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Once we have a regression line y=ax+b, we can use it to predict y by plugging in a given value of x.
For example, suppose we have data on the body temperature of babies in the first 12 months after birth. Let's say we have the regression line between body temperature T in °C at age M months:
We can then predict the body temperature of a 10 month old baby by plugging in M=10:
It is also possible too use a regression line y=ax+b to predict x using a given value of y:
But this is not always going to be a reliable process, since the best fit line is determined (via regression) so as to minimize the difference between the real y's and the predicted y's, and so the difference in x could be way off:
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When using a regression line to predict y from x, we need to be aware of the danger of extrapolation. This occurs when we try to predict y for a value of x far outside the range of x values in our data. For such an x, we cannot trust that the relationship is the same.
In our earlier example, we had data about the body temperature of babies in the first 12 months after birth. We had the regression equation:
It would be a foolish extrapolation to use this formula to predict the body temperature of an 80 year old... 80 years is 960 months, so
(a block of ice).
Best fit lines can also be drawn approximately by eye. We start by finding the average x and y, giving the point (xˉ,yˉ). We then take a rule and place it on this point, and adjust the slope until we find a reasonable best fit line.
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In the same way that we can plot a straight line minimizing the vertical distances from points (x,y), we can plot a straight line minimizing the horizontal distances. This is called an x on y regression line.
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With this line we can make reliable predictions of x given y, so long as we are not extrpolating.
Example
Find the x on y regression line for the following data, and hence predict x when y=14.
Using a calculator (putting the y-values in the Xlist and x-values in the Ylist) we find
Plugging in y=14, we find x=4.72 (or 4.73 is you use exact values).