Statistics Toolkit (DP IB Applications & Interpretation (AI))

False.

Forming a sample by including every 10th person from a list is not an example of stratified sampling.

It is an example of systematic sampling.

The main difference between stratified sampling and quota sampling is how members for each group are selected.

For stratified sampling, simple random sampling is used on each group of the population. Whereas for quota sampling, the members do not have to be selected randomly, e.g. convenience sampling could be used.

To calculate the number of members of each group to include in a stratified sample use the formula:

If there are 100 animals on a farm (60 chickens and 40 sheep), then to find the number of chickens to include in a stratified sample of 30 animals you would calculate:

The mode of a data set is the item(s) that occurs the most often.

False.

Not all data sets have exactly one mode.

A data set may have:

• no mode

• more than one mode.

To find the median of ungrouped data:

• put the data in order from smallest to largest,

• find the middle value(s).

If there are two middle values then find the midpoint of them.

For example, the median of 1, 2, 3, 4 is 2.5 (i.e. the midpoint of 2 and 3).

To find the mean of ungrouped data:

• add all the values together,

• divide by the number of values.

For example, the mean of 1, 2, 3, 4 can be found by calculating:

This formula is used to calculate the mean of a data set. This is denoted as .

True.

The lower quartile of a set of data splits the lowest 25% from the highest 75%.

For example, the lower quartile of 1, 2, 3, 4 is 1.5.

The interquartile range is the range of the central 50% of data.

It can be found by subtracting the lower quartile from the upper quartile.

False.

The range is the difference between the lowest value and the highest value.

The range of 2, 3, 1, 5, 8 is 8 - 1 = 7.

False.

The standard deviation is not a measure of central tendency.

The standard deviation is a measure of dispersion, it measures how spread out the data is about the mean.

The standard deviation is the positive square-root of the variance.

(Equivalently, the variance is the standard deviation squared.)

denotes the population variance.

To calculate the mean from a frequency table containing ungrouped data:

• multiply each value (x_i) by its frequency (f_i),

• add these products together,

• divide by the total frequency (n).

This can be written as the formula .

False.

The mode is the value with the highest frequency. Therefore the modal score is 5.

True.

It is possible to find the exact median from a frequency table of ungrouped data.

You can find the median by finding the middle value. You can use cumulative frequency to help to find which value is the middle value.

You can also use your GDC to find the median.

To calculate the mean from a frequency table containing ungrouped data:

• find the mid-interval value (midpoint) of each group (x_i),

• multiply each midpoint by its frequency (f_i),

• add these products together,

• divide by the total frequency (n).

This can be written as the formula .

It is not possible to calculate the exact mean of grouped data because the exact individual values are unknown.

To calculate the mid-interval value of a group in a frequency table of grouped data:

• add the upper and lower boundaries of the group,

• divide by 2.

For example, the mid-interval value of the group can be found by calculating .

True.

If you add 5 to each value in a data set, then the mean also increases by 5.

is the mean of a data set. The constant is subtracted from each value in the data set. The mean of the new values is .

If you subtract a constant from all the values in the data set then you also subtract it from the mean of the original values to find the mean of the new values.

If each value in a data set is doubled, then the mean is also doubled.

is the mean of a data set. Each value in the data set is multiplied by the constant . The mean of the new values is .

If you multiply all the values in the data set by a constant then you also multiply the mean of the original values by the constant to find the mean of the new values.

False.

If you add 5 to each value in a data set, then the standard deviation does not change.

Adding or subtracting a constant to the values in a data set does not affect the standard deviation.

is the variance of a data set. Each value in the data set is multiplied by the constant . The variance of the new values is .

If you multiply all the values in the data set by a constant then you multiply the variance of the original values by the constant squared to find the variance of the new values.

False.

If you multiply each value in a data set by -2, then the standard deviation is multiplied by 2.

The standard deviation is always positive.

Outliers are extreme data values that do not fit with the rest of the data.

The boundaries for outliers in a data set are found by:

• multiplying the interquartile range by 1.5,

• subtracting this from the lower quartile and adding it to the upper quartile.

It can be written as a formula. x is an outlier if x < Q₁ - 1.5 × IQR or x > Q₃ + 1.5 × IQR.

False.

Not all outliers are errors.

Outliers should be removed from a data set if they are clearly errors. However, if they are possibly valid values then they should be included.

A cross (×) is used to represent an outlier on a box and whisker diagram.

The five values needed to draw a box and whisker diagram are: lowest data value, lower quartile, median, upper quartile, and highest data value.

The box in a box and whisker diagram represents the central 50% of the data set.

The whiskers represent the lowest 25% and the highest 25% of the data.

False.

To draw a cumulative frequency diagram, you do not plot the midpoint of a group against its frequency. This is a mistake that students often make on the exam.

You plot the endpoint of a group against its cumulative frequency.

If the total frequency is 100, you can use a cumulative frequency diagram to estimate the median value.

1. Draw a horizontal line from 50 () on the vertical axis to the curve.

2. Draw a vertical line down from this point on the curve to the horizontal axis.

The value on the horizontal axis is an estimate of the median.

To estimate the lower quartile using a cumulative frequency diagram:

1. Divide the total frequency by 4.

2. Draw a horizontal line from that result on the vertical axis to the curve.

3. Draw a vertical line down from this point to the horizontal axis.

The number on the horizontal axis is an estimate for the lower quartile.

A frequency histogram clearly shows the frequency of class intervals for grouped data with equal class intervals.

False.

You do not leave a gap between the bars when drawing a histogram.

A box and whisker diagram shows the median.

When working with grouped data, a histogram clearly shows the modal group.

The mean is most affected by outliers.

The mode is the only measure of central tendency (mode, median, mean) that can be applied to qualitative data.

False.

When comparing two data sets, the one with the higher mean is not always better. It depends on the context.

If comparing times to complete a race, the smaller mean is better.

If comparing scores, the bigger mean is better.

True.

The interquartile range is not affected by outliers.

The interquartile range only uses at the central 50% of the data.

When comparing two data sets, the one with the bigger interquartile range is more spread out about the median.

If you are comparing data sets which contain outliers, you should use the interquartile range as this is not affected by outliers.