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What is the modal range of the students? Find the mean and median age. Construct a frequency table for the data. Use your table to calculate the mean & median. State what is the mode of the distribution?

Mathematics

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Answers

2 parts
ii
17.11
15
iii

12 and 26\text{12 and 26}

Step-by-step solution

2 steps

Here's the solution for question 4:

Part i

Construct a frequency table for the data.

First, we list the scores and count their frequencies. The scores are:
12, 26, 15, 14, 14, 12, 16, 26, 12, 15, 15, 14, 26, 26, 15, 14, 19, 12, 26, 19, 19, 19, 14, 15, 26, 12, 14, 26, 12, 15, 26, 12

Let xx be the score and ff be the frequency. We also calculate fxfx and the cumulative frequency (CF).

Score (xx)Frequency (ff)fxfxCumulative Frequency (CF)
128968
1479815
15710522
1611623
1947627
26820835
Total35599

Part ii17.11

Use your table to calculate the mean and median.

To calculate the mean (xˉ\bar{x}), we use the formula xˉ=∑fxN\bar{x} = \frac{\sum fx}{N}, where NN is the total number of students.
From the table, ∑fx=599\sum fx = 599 and N=35N = 35.

  1. Calculate the mean.

    xˉ=∑fxN\bar{x} = \frac{\sum fx}{N} xˉ=59935\bar{x} = \frac{599}{35} xˉ≈17.114\bar{x} \approx 17.114

    The mean score is 17.11\boxed{17.11} (rounded to two decimal places).

  2. Calculate the median.

    The total number of data points is N=35N = 35. The position of the median is given by N+12\frac{N+1}{2}.

    Median position=35+12=362=18th\text{Median position} = \frac{35+1}{2} = \frac{36}{2} = 18^{\text{th}}

    We look at the cumulative frequency column to find the score at the 18th position.

    • The first 8 scores are 12.
    • The scores from 9 to 15 are 14.
    • The scores from 16 to 22 are 15. Since the 18th position falls within the range of scores that are 15, the median is 15. The median score is 15\boxed{15}.

Part iii

12 and 26\text{12 and 26}

State what is the mode of the distribution?

The mode is the score that appears most frequently in the data.
From the frequency table:

  • Score 12 has a frequency of 8.
  • Score 14 has a frequency of 7.
  • Score 15 has a frequency of 7.
  • Score 16 has a frequency of 1.
  • Score 19 has a frequency of 4.
  • Score 26 has a frequency of 8. The highest frequency is 8, which corresponds to two scores: 12 and 26. The distribution is bimodal. The mode of the distribution is 12 and 26\boxed{\text{12 and 26}}.
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