TSI Math Study Guide
Center and Spread
Statistics helps us describe and compare sets of data. On the TSI, you may need to calculate or interpret mean, median, mode, and range, and understand measures of spread such as interquartile range and standard deviation.
Measures of center
Describe a typical or central value.
mean · median · mode
Measures of spread
Describe how much the values vary.
range · IQR · standard deviation
Mean
Mean = sum of all valuesnumber of values
Working backward from the mean
The shortcut
Total = (Mean)(Number of values)
Median
Odd number of values
Five values — the middle one is the third.Median = 9
Even number of values
Six values — average the two middle ones.Median = 10 + 142 = 12
Mode and range
Mode — the most frequent value
Mode = 7
A dataset can have more than one mode, or no mode at all if nothing repeats.
Range — maximum minus minimum
Range = 21 − 6 = 15
A quick measure of spread, but it only looks at the two extremes.
How outliers affect the mean and median
No outlier
Last value replaced with 100
The median didn’t move. The mean was pulled sharply upward by one value.
The idea this illustrates
The mean is more sensitive to outliers than the median. When a dataset has an extreme value, the median may describe the typical value better.
Quartiles and interquartile range
Lower half: 2, 4, 6, 8
Q1 = 4 + 62 = 5
Upper half: 10, 12, 14, 16
Q3 = 12 + 142 = 13
Unlike the range, the IQR isn’t determined by the minimum and maximum, so it’s less sensitive to extreme values. It’s especially useful for reading box plots, which return in Data Interpretation.
Standard deviation
Dataset A — mean 50
Clustered close to 50 → small standard deviation.
Dataset B — mean 50
Spread far from 50 → large standard deviation.
Comparing datasets
Two classes both average 80 on a test. Class A has a standard deviation of 3; Class B has a standard deviation of 12. The centers are the same, but Class A’s scores cluster tightly around 80 while Class B’s vary widely.
The takeaway
Knowing the mean alone doesn’t give a complete picture of a dataset.
Choosing the appropriate measure
Mean
An average that incorporates every value.
Median
The middle value — especially when extreme values might distort the mean.
Mode
When the most frequent value matters.
Range
A quick overall spread from minimum to maximum.
IQR
The spread of the middle half of the data.
Standard deviation
How closely values cluster around the mean.
A TSI question may give you several statistics and ask what they reveal, rather than asking you to calculate them.
TSI strategy: put the data in order
With a small dataset, arrange the numbers from least to greatest first. That immediately reveals:
For mean questions, check whether you actually need to add every value. If you’re given the mean and the count, use Total = (Mean)(Number of values) — especially for missing-value problems.
Center
mean, median, mode
Spread
range, IQR, standard deviation
Two datasets can have exactly the same mean and still look very different, because one has much greater spread.