IB Mathematics Analysis and Approaches SL 4.3—mean, Median, Mode. Mean of Grouped Data, Standard Deviation. Quartiles, IQR Notes

One Number To Stand For Many: Center And Spread

1. Suppose two classes both average 60% on a test, so on paper they look identical.
2. In one class almost everyone scored near 60, but in the other, half scored 30 and half scored 90.
3. To describe a data set honestly you need two kinds of number.
   1. A measure of central tendency says where the middle of the data sits.
   2. A measure of dispersion says how spread out the values are around that middle.

Mean, Median, And Mode: Three Kinds Of Average

1. Mean
   1. The mean is the arithmetic average: add every value and divide by how many there are.
2. Median
   1. The median is the middle value once the data is placed in ascending order.
   2. With an even number of values, take the mean of the two middle values.
3. Mode
   1. The mode is the value that occurs most often.
   2. A data set can have one mode, several modes, or no mode at all.
4. For the small set $4, 7, 9, 10, 12$ the mean is $\dfrac{42}{5}=8.4$, the median is the middle value $9$, and there is no mode.

Why Skew And Outliers Split The Mean From The Median

1. An outlier is a value far from the rest of the data, and skew is a long tail stretching to one side.
2. The mean adds the outlier's full size into the total, so a single extreme value drags it toward the tail.
3. The median only cares about position in the ordered list, so one extreme value barely moves it.

Estimating The Mean From Grouped Data

1. Data often arrives already sorted into class intervals, for example commute times grouped as $0$ to $10$ minutes, $10$ to $20$ minutes, and so on.
2. To estimate the mean, treat every value in a class as sitting at the class mid-interval value, the midpoint of the interval.

A Fully Worked Grouped Mean

1. Take the frequency table of commute times above.
2. So the estimated mean is: $$\bar{x}=\dfrac{1050}{40}=26.25 \text{ minutes}$$
3. It is an estimate because we assumed every value sat exactly at its midpoint, which is only approximately true.

The Modal Class

1. For grouped data you cannot give a single mode, so you give the modal class instead.
2. The modal class is the class interval with the highest frequency.
3. In the commute table the $20$ to $30$ class has frequency $12$, the largest, so it is the modal class.
4. The modal class is only meaningful when all classes have equal width.
5. If the widths differ, a wide class can collect more values just by being wider, which makes the raw frequency misleading.

Measuring Spread: Range, IQR, And Standard Deviation

The two data sets $4,5,6,7,8$ and $1,3,6,9,11$ both have mean $6$, yet the second is clearly more spread out.

Range And Interquartile Range

1. The range is the largest value minus the smallest value.
   1. It is quick but fragile, since it depends only on the two most extreme values.
2. Quartiles cut an ordered data set into four equal parts.
   1. $Q_1$, the lower quartile, sits a quarter of the way through the ordered data.
   2. $Q_3$, the upper quartile, sits three quarters of the way through.
3. The interquartile range is $\text{IQR}=Q_3-Q_1$, the spread of the middle $50\%$ of the data.

Standard Deviation: Typical Distance From The Mean

1. The standard deviation $\sigma$ measures how far, on average, the values lie from the mean.
   1. A small $\sigma$ means the data huddles close to the mean.
2. To see the idea, take $4,5,6,7,8$ with mean $6$: the distances from the mean are $2,1,0,1,2$.
   1. Squaring, averaging, then square-rooting gives $\sqrt{\dfrac{4+1+0+1+4}{5}}=\sqrt{2}\approx 1.41$.
3. These hand steps are only to build intuition, not an exam method.

Variance Is The Square Of The Standard Deviation

1. The variance is simply $\sigma^2$, the square of the standard deviation.
2. If the GDC gives $\sigma=3.69$, the variance is $\sigma^2\approx 13.6$.

Shifting And Scaling: How Transforms Change Mean And SD

Sometimes every value in a data set is changed by the same rule, for example converting marks or shifting a baseline.

Adding A Constant Shifts The Mean But Not The Spread

1. If you add a constant $c$ to every value, the new mean is the old mean plus $c$.
2. Every value moves the same distance in the same direction, so the whole data set slides along without stretching.
3. Because the gaps between values are unchanged, the standard deviation and variance stay exactly the same.
4. Here is why: each distance from the mean is $(x_i+c)-(\bar{x}+c)=x_i-\bar{x}$, identical to before.

Multiplying By A Constant Scales Both Mean And Spread

1. If you multiply every value by a constant $k$, the new mean is $k$ times the old mean.
2. The standard deviation is multiplied by $|k|$, and the variance by $k^2$.
3. Here is why: each distance becomes $k x_i - k\bar{x}=k(x_i-\bar{x})$, so every deviation is stretched by $k$.
4. Standard deviation, built from those deviations, therefore scales by $|k|$.