## Frequency Histograms

Questions 1 to 5 relate to the following graph.

Figure 1 - frequency histogram of resting heart rate in HP3 students: The frequency distribution of resting heart rate in HP3 students is best described as:

Normal (Bell-shaped / Gausssian)

Left skewed

Right skewed

Bimodal

None of the above

### 2

The mode for the frequency distribution of resting heart rate in HP3 students is approximately:

100 bpm

40 bpm

70 bpm

50 bpm

None of the above

Mode is the most frequent value and therefore the heart rate value at which the frequency histogram is at its peak.

### 3

Regarding Figure 1, which if the following is correct?

Mode > Median > Mean

Mode ≈ Median ≈ Mean

Mean > Median > Mode

Median > Mean > mode

None of the above

In a bell-shaped (normal) distribution, Mode ≈ Median ≈ Mean (i.e. all three of these measures of central tendency are approximately the same).

### 4

Regarding Figure 1, which if the following is correct?

2.5% of the students will have a resting heart rate greater than two standard deviations above the mean

2.5% of the students will have a resting heart rate less than two standard deviations below the mean

50% of the students will have a resting heart rate greater than 70 bpm

50% of the students will have a resting heart rate less than 70 bpm

All of the above

For a and b, see Figure 2. For c and d, remember that median is the middle value, meaning that 50% of values will be greater than median and 50% of values will be less than median. In a bell-shaped (normal) distribution, Mode ≈ Median ≈ Mean. Here mode is 70 and therefore median is also around 70. Figure 2: ### 5

If the standard deviation for the frequency distribution (Figure 1) is 11 bpm, then which of the following statements is correct?

Approx two-thirds (68%) of the students will have resting HR between 59 and 81 bpm

Almost 14% of the students will have resting HR between 48 and 59 bpm

Almost 14% of the students will have resting HR between 81 and 92 bpm

a, b and c

None of the above

In a bell-shaped (normal) distribution, Mode ≈ Median ≈ Mean. Here mode is 70 and therefore mean is also around 70. Mean minus 1 standard deviation is 59. Mean plus one standard deviation is 81. Mean minus two standard deviations is 70 - (2 X 11)=48. Mean plus two standard deviations is 70 + (2 X 11)= 92. See Figure 2.

Figure 3 - Mean, median and mode in a skewed distribution: ## Box Plots

The following information and Figure 4 relate to questions 6 to 10.

The time from diagnosis to death in 100 patients with an aggressive form of pancreatic cancer is depicted as a box plot.

Figure 4 - Box plot of time from diagnosis to death in 100 patients with pancreatic cancer: Regarding box plots, which one of the following statements is correct?

Box plots are never used for normally distributed data

Box plots are only used for right skewed data

Box plots are only used for left skewed data

Box plots are usually used to depict the distribution of skewed data, especially where there are ‘outliers’

None of the above

Box plots are usually used to depict the spread of skewed data (right or left skew), especially where there are outliers and extreme outliers. However, box plots may also be used to depict the spread of normally distributed data.

### 7

Excluding outliers, the minimum and maximum survival times among these patients are:

12 and 24 months, respectively

1.5 and 16 months, respectively

7.5 and 11.5 months, respectively

10 and 16 months, respectively

None of the above

The ends of the ‘whiskers’ on a box plot are the minimum and maximum values (excluding outliers - circles, and extreme outliers - stars).

### 8

Excluding outliers, the first (Q1) and third (Q3) quartiles for survival times are:

12 and 24 months, respectively

1.5 and 16 months, respectively

7.5 and 11.5 months, respectively

10 and 16 months, respectively

None of the above

The bottom and the top of the ‘box’ denote the first and third quartile values.

### 9

Which of the following statements is correct?

Half (50%) of the patients will have a survival time between 12 and 24 months

Half (50%) of the patients will have a survival time between 1.5 and 16 months

Half (50%) of the patients will have a survival time between 7.5 and 11.5 months

Half (50%) of the patients will have a survival time between 10 and 16 months

None of the above

25% of patients will be in each quartile. Therefore 50% will be between Q1 and Q3 (‘interquartile range’).

### 10

For the box plot in figure 4, what is the inter-quartile range?

6 months

14.5 months

4 months

6 months

None of the above

Interquartile range = Q3 - Q1 = 11.5 - 7.5 = 4 months.

More explanation about box plots and quartiles follows this in revision lecture slides.