Cumulative frequency/box plots

1. What is Cumulative Frequency?

Cumulative Frequency is a way of calculating a running total of frequencies as you move through the data. The word "cumulative" means "adding up," and "frequency" refers to how often something happens. So, cumulative frequency tells us how many data points are below a certain value.

2. Adding a Cumulative Frequency Column

To create a cumulative frequency column, we add up the frequencies as we move through the table. The cumulative frequency is the running total of the frequencies.

---

3. Drawing the Cumulative Frequency Curve

The cumulative frequency curve (also known as an ogive) is a graph that shows the running total of frequencies, helping us visualise how data accumulates over time or intervals.

Steps for Drawing a Cumulative Frequency Curve:

1. Create a cumulative frequency column in your table (add up the frequencies as you go).
2. Plot the cumulative frequency ($y-axis$) against the upper boundary of each group ($x-axis)$.
3. Once all points are plotted, join them up with a smooth curve.
4. Make sure your graph starts at ($0, 0$), as no data occurs below $0$.
5. Label the axes correctly—the $x-axis$ is the range of data (e.g., time spent playing), and the $y-axis$ is cumulative frequency.

---

4. Interpreting the Cumulative Frequency Curve

Once the cumulative frequency curve is drawn, it allows us to estimate key statistics such as the median, lower quartile ($Q1$), upper quartile ($Q3$), and the interquartile range ($IQR$).

Estimating Quartiles:

1. Median: This is the 50th percentile, or halfway point. To find it, go to half the total frequency on the $y-axis$ and read off the corresponding value on the $x-axis$.

• In this example, the total frequency is $40$, so the median is at the 20th data point.

2. Lower Quartile ($Q1$): This is the 25th percentile. Find 25% of the total frequency, and read off the corresponding value on the $x-axis$.

• For a total frequency of $40$, find $Q1$ at the 10th data point.

3. Upper Quartile ($Q3$): This is the 75th percentile. Find 75% of the total frequency and read off the corresponding value on the $x-axis$.

• For a total frequency of $40$, $Q3$ is at the 30th data point.

4. Interquartile Range ($IQR$): This measures the spread of the middle 50% of the data and is calculated as:

$$IQR=Q3−Q1$$

---

5. Estimating the Median and Quartiles from a Cumulative Frequency Curve

(a) Finding the Median

The median is the value that splits the data into two equal halves. For grouped data, the median can be estimated from the cumulative frequency curve.

Steps to Find the Median:

5. Identify 50% of the total frequency (this is half of the data points).

• In this example, the total frequency is $40$, so 50% is 20.

6. Draw a horizontal line from the 20th cumulative frequency point on the y-axis until it intersects the cumulative frequency curve.
7. From the point where the horizontal line meets the curve, draw a vertical line down to the $x-axis$.
8. The point where the line touches the x-axis is the estimated median.

(b) Finding the Quartiles

The lower quartile ($Q1$) is the value below which 25% of the data lies, and the upper quartile ($Q3$) is the value below which 75% of the data lies. The interquartile range ($IQR$) is the difference between $Q3$ and $Q1$, showing the spread of the middle 50% of the data.

Steps to Find the Quartiles:

9. Lower Quartile ($Q1$):

• Find 25% of the total frequency (for a total of $40, Q1$ is at the 10th data point).
• Draw a horizontal line from the 10th cumulative frequency point to the curve and then draw a vertical line down to the $x-axis$.
• The value on the $x-axis$ is the lower quartile.

10. Upper Quartile ($Q3$):

• Find 75% of the total frequency (for a total of $40$, $Q3$ is at the 30th data point).
• Draw a horizontal line from the 30th cumulative frequency point to the curve and then draw a vertical line down to the $x-axis$.
• The value on the $x-axis$ is the upper quartile.

6. Interquartile Range ($IQR$)

The interquartile range ($IQR$) is the difference between the upper and lower quartiles:

$$IQR=Q3−Q1$$

The $IQR$ tells us how spread out the middle 50% of the data is, and it is a useful measure because it is not affected by outliers.

---

7. Drawing Box Plots from a Cumulative Frequency Curve

A box plot (or box-and-whisker plot) visually summarises data by showing the median, quartiles, and range. Box plots provide a clear representation of the spread of the data.

Steps for Drawing a Box Plot:

1. Identify the five-number summary
2. Draw a number line
3. Draw the box
4. Draw the whiskers

---

5. Identify the five-number summary:

• Minimum value: The smallest value in the data set.
• Lower Quartile ($Q1$): The value below which 25% of the data lies.
• Median: The middle value (50% of the data lies below this).
• Upper Quartile ($Q3$): The value below which 75% of the data lies.
• Maximum value: The largest value in the data set.

---

6. Draw a number line: Mark the minimum value, $Q1$, median, $Q3$, and maximum value on the number line.

---

7. Draw the box: The box extends from $Q1$ to $Q3$ with a line inside the box at the median.

---

8. Draw the whiskers: These lines extend from the box to the minimum and maximum values.

---

Interpreting the Box Plot

The box plot clearly summarises the data by showing:

• The spread of the middle 50% of the data (between $Q1$ and $Q3$) through the interquartile range ($IQR$).
• The overall range of the data (from the minimum to the maximum values).
• The median, showing the central value.

Key Points to Note:

1. Range:

• The range is the difference between the maximum and minimum values:

$$Range=Maximum−Minimum=10−0=10 hours$$

2. Interquartile Range ($IQR$):

• The $IQR$ is the spread of the middle 50% of the data:

$$IQR=Q3−Q1=3.8−2.6=1.2 hours$$

• The $IQR$ tells us how spread out the central data values are. In this case, the middle 50% of the students played video games for a period of 1.2 $hours$.