Math / Data & Statistics

QuestionThe table gives information about the times taken by 80 people to run a race.
Time taken ( $t$ minutes) Cumulative Frequency $\begin{array}{ll} 50<t \leq 60 & 15 \\ 50<t \leq 70 & 31 \\ 50<t \leq 80 & 52 \\ 50<t \leq 90 & 66 \\ 50<t \leq 100 & 74 \\ 50<t \leq 110 & 80 \end{array}$
This information is shown on the cumulative frequency graph below.
Use this graph to find an estimate for the interquartile range of the times taken.

Studdy Solution

STEP 1

What is this asking? Find the range between the time it takes for the fastest 25% and the slowest 25% of people to finish the race. Watch out! Don't confuse cumulative frequency with regular frequency.
Remember, cumulative frequency adds up as you go!

STEP 2

1. Find the first quartile (Q1)
2. Find the third quartile (Q3)
3. Calculate the interquartile range (IQR)

STEP 3

The first quartile, or $Q_1$ , is the value below which 25% of the data falls.
Since there are 80 people, we need to find the cumulative frequency that corresponds to 25% of 80.

STEP 4

Calculate the position for $Q_1$ : $Q_1 \text{ position} = 0.25 \cdot 80 = 20$

STEP 5

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 20.
This is your $Q_1$ .

STEP 6

The third quartile, or $Q_3$ , is the value below which 75% of the data falls.
Again, with 80 people, find the cumulative frequency that corresponds to 75% of 80.

STEP 7

Calculate the position for $Q_3$ : $Q_3 \text{ position} = 0.75 \cdot 80 = 60$

STEP 8

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 60.
This is your $Q_3$ .

STEP 9

The interquartile range is the difference between the third quartile and the first quartile.
This tells us the spread of the middle 50% of the data.

STEP 10

Calculate the IQR: $\text{IQR} = Q_3 - Q_1$

STEP 11

The interquartile range (IQR) is the difference between the time at the 60th position and the time at the 20th position on the cumulative frequency graph.
This gives us the spread of the middle 50% of the race times.

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Studdy Solution

STEP 1

What is this asking? Find the range between the time it takes for the fastest 25% and the slowest 25% of people to finish the race. Watch out! Don't confuse cumulative frequency with regular frequency.
Remember, cumulative frequency adds up as you go!

STEP 2

1. Find the first quartile (Q1)
2. Find the third quartile (Q3)
3. Calculate the interquartile range (IQR)

STEP 3

The first quartile, or $Q_1$ , is the value below which 25% of the data falls.
Since there are 80 people, we need to find the cumulative frequency that corresponds to 25% of 80.

STEP 4

Calculate the position for $Q_1$ : $Q_1 \text{ position} = 0.25 \cdot 80 = 20$

STEP 5

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 20.
This is your $Q_1$ .

STEP 6

The third quartile, or $Q_3$ , is the value below which 75% of the data falls.
Again, with 80 people, find the cumulative frequency that corresponds to 75% of 80.

STEP 7

Calculate the position for $Q_3$ : $Q_3 \text{ position} = 0.75 \cdot 80 = 60$

STEP 8

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 60.
This is your $Q_3$ .

STEP 9

The interquartile range is the difference between the third quartile and the first quartile.
This tells us the spread of the middle 50% of the data.

STEP 10

Calculate the IQR: $\text{IQR} = Q_3 - Q_1$

STEP 11

The interquartile range (IQR) is the difference between the time at the 60th position and the time at the 20th position on the cumulative frequency graph.
This gives us the spread of the middle 50% of the race times.

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Studdy Solution

STEP 1

What is this asking? Find the range between the time it takes for the fastest 25% and the slowest 25% of people to finish the race. Watch out! Don't confuse cumulative frequency with regular frequency.Remember, cumulative frequency adds up as you go!

STEP 2

1. Find the first quartile (Q1)2. Find the third quartile (Q3)3. Calculate the interquartile range (IQR)

STEP 3

The first quartile, or Q1 Q_1 Q1​, is the value below which 25% of the data falls.Since there are 80 people, we need to find the cumulative frequency that corresponds to 25% of 80.

STEP 4

**Calculate** the position for Q1 Q_1 Q1​: Q1 position=0.25⋅80=20 Q_1 \text{ position} = 0.25 \cdot 80 = 20 Q1​ position=0.25⋅80=20

STEP 5

**Locate** the position on the cumulative frequency graph.Look for the time that corresponds to a cumulative frequency of **20**.This is your Q1 Q_1 Q1​.

STEP 6

The third quartile, or Q3 Q_3 Q3​, is the value below which 75% of the data falls.Again, with 80 people, find the cumulative frequency that corresponds to 75% of 80.

STEP 7

**Calculate** the position for Q3 Q_3 Q3​: Q3 position=0.75⋅80=60 Q_3 \text{ position} = 0.75 \cdot 80 = 60 Q3​ position=0.75⋅80=60

STEP 8

**Locate** the position on the cumulative frequency graph.Look for the time that corresponds to a cumulative frequency of **60**.This is your Q3 Q_3 Q3​.

STEP 9

The interquartile range is the difference between the third quartile and the first quartile.This tells us the spread of the middle 50% of the data.

STEP 10

**Calculate** the IQR: IQR=Q3−Q1 \text{IQR} = Q_3 - Q_1 IQR=Q3​−Q1​

STEP 11

The interquartile range (IQR) is the difference between the time at the 60th position and the time at the 20th position on the cumulative frequency graph.This gives us the spread of the middle 50% of the race times.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Find the range between the time it takes for the fastest 25% and the slowest 25% of people to finish the race. Watch out! Don't confuse cumulative frequency with regular frequency.
Remember, cumulative frequency adds up as you go!

1. Find the first quartile (Q1)
2. Find the third quartile (Q3)
3. Calculate the interquartile range (IQR)

The first quartile, or $Q_1$ , is the value below which 25% of the data falls.
Since there are 80 people, we need to find the cumulative frequency that corresponds to 25% of 80.

Calculate the position for $Q_1$ : $Q_1 \text{ position} = 0.25 \cdot 80 = 20$

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 20.
This is your $Q_1$ .

The third quartile, or $Q_3$ , is the value below which 75% of the data falls.
Again, with 80 people, find the cumulative frequency that corresponds to 75% of 80.

Calculate the position for $Q_3$ : $Q_3 \text{ position} = 0.75 \cdot 80 = 60$

Locate the position on the cumulative frequency graph.
Look for the time that corresponds to a cumulative frequency of 60.
This is your $Q_3$ .

The interquartile range is the difference between the third quartile and the first quartile.
This tells us the spread of the middle 50% of the data.

Calculate the IQR: $\text{IQR} = Q_3 - Q_1$

The interquartile range (IQR) is the difference between the time at the 60th position and the time at the 20th position on the cumulative frequency graph.
This gives us the spread of the middle 50% of the race times.