Averages And Range

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Notes

Discrete & Continuous Data

**Continuous data** can take any numerical value on a scale (e.g., height, weight, time).
**Discrete data** can only take particular values, often integers (e.g., number of people, shoe sizes).
If you need a scale to measure it, it is likely continuous.

Mean, Median & Mode

**Mode**: value that appears most often; can be more than one mode.
**Median**: middle value when data is ordered; for even number of values, take midpoint of the two middle values.
**Mean**: sum of values divided by number of values; can be a decimal.
Use median when data has extreme values (outliers); avoid mode if more than one mode; mode is the only average for non-numerical data.

Calculations with the Mean

Mean = total of values ÷ number of values.
Rearrange: total of values = mean × number of values.
Use totals before and after to find missing values or effect of adding/removing a value.

Averages from Tables

**Mode**: data value with highest frequency.
**Median**: the ((n+1)/2)th value; use cumulative frequency to locate it.
**Mean**: total of (data value $\times frequency)$ divided by total frequency.
**Range**: largest data value minus smallest data value (not difference of frequencies).

Averages from Grouped Data

Grouped data uses class intervals; exact mean cannot be found, only estimated.
**Estimated mean**: use midpoint of each interval as representative value, then total of $(midpoint \times frequency)$ divided by total frequency.
**Modal class**: class interval with highest frequency.
**Median class**: class interval containing the ((n+1)/2)th value.

Range & Interquartile Range

**Range** = highest value – lowest value; measures spread but affected by outliers.
**Lower quartile (LQ)** is median of lower half of data; **upper quartile (UQ)** is median of upper half.
**Interquartile range (IQR)** $= UQ$ – LQ; measures spread of middle 50% and is not affected by outliers.

Comparing Data Sets

Compare averages (mode, median, mean) and spreads (range, IQR).
Write conclusions: state which is higher/lower and what that means in context (e.g., 'on average, slugs travel further').
Use phrases like 'more consistent', 'less variation' for smaller spread.
Be aware of limitations: small sample size, bias, or influence of outliers.

Mean, Median, Mode on a Number Line

Quartiles and IQR

Estimating Mean from Grouped Data

Practice questions

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1.State whether the following data is discrete or continuous: The weights of dogs participating in a dog show.
Easy
- ADiscrete
- BContinuous
- CNeither
- DBoth
2.Find the range of the shoe sizes: 6, 7, 8, 9, 10, 11, 12.
Easy
- A6
- B7
- C5
- D12
3.For the data set: 1, 2, 2, 5, 5, 6, what is the mode?
Easy
- A2
- B5
- C2 and 5
- DNo mode
4.Find the median of the numbers: 4, 2, 3.
Easy
- A3
- B2
- C4
- D2.5
5.Calculate the mean of: 1, 2, 6.
Easy
- A3
- B4
- C2
- D3.5
6.The students in Class A and Class B take the same examination. The lowest score in Class A is 39 and the range is 57. The lowest score in Class B is 33 and the range is 60. Find the range of scores for all the students in both classes.
Medium
- A117
- B57
- C60
- D87
7.Roberto records the value of each of his coins. The table shows: Value (cents): 1,2,5,10,20,50; Frequency: 3,1,3,2,4,2. Find the range.
Medium
- A49 cents
- B50 cents
- C51 cents
- D1 cent
8.For the same coin data, find the mode.
Medium
- A1 cent
- B2 cents
- C20 cents
- D50 cents

Summary

## Averages & Range - Quick Revision ### Key Definitions - **Mean** = sum of values ÷ number of values. Use formula: total = mean × number. - **Median** = middle value when data is ordered. For n values, median is the ((n+1)/2)th value. - **Mode** = most frequent value. - **Range** = highest value - lowest value. - **Interquartile Range (IQR)** = Upper Quartile - Lower Quartile. Measures spread of middle 50%. ### Discrete vs Continuous Data - **Discrete**: specific values (e.g., shoe sizes, number of pets). - **Continuous**: any value on a scale (e.g., height, weight, time). ### Averages from Frequency Tables - **Mode**: value with highest frequency. - **Median**: find cumulative frequencies; locate ((n+1)/2)th value. - **Mean**: add column 'x × f', sum it, divide by total frequency. - **Range**: largest data value - smallest data value (not frequencies). ### Averages from Grouped Data - **Estimated Mean**: use midpoints of intervals. Formula: Σ(midpoint × frequency) ÷ total frequency. - **Modal Class**: class interval with highest frequency. - **Median Class**: find cumulative frequencies; locate ((n+1)/2)th value's interval. ### Comparing Data Sets - Compare **averages** (median/mean) and **spreads** (range/IQR). - Write: "[Group A] has a higher/lower [average] than [Group B], meaning ..." - Write: "[Group A] has a smaller/larger [spread] than [Group B], meaning ..." ### Important Tips - Mean is affected by outliers; median is better for skewed data. - For even number of values, median is midpoint of two middle values. - Quartiles: LQ = median of lower half (exclude median if odd n); UQ = median of upper half. - IQR = UQ - LQ; not affected by outliers.

Key terms

Mean: The sum of all values divided by the number of values. Formula: mean = total of values ÷ number of values.
Median: The middle value when data is arranged in order. For n values, it is the ((n+1)/2)th value.
Mode: The value that appears most often in a data set.
Range: The difference between the highest and lowest values. Range = highest - lowest.
Interquartile Range (IQR): The difference between the upper quartile and the lower quartile. IQR = UQ - LQ. Measures spread of middle 50% of data.
Lower Quartile (LQ): The median of the lower half of the data. 25% of data lies below this value.
Upper Quartile (UQ): The median of the upper half of the data. 75% of data lies below this value.
Discrete Data: Numerical data that can only take specific values (often integers). E.g., number of pets, shoe sizes.
Continuous Data: Numerical data that can take any value within a range. E.g., height, weight, time.
Modal Class: For grouped data, the class interval with the highest frequency.
Estimated Mean: An approximation of the mean for grouped data, calculated using midpoints: Σ(midpoint × frequency) ÷ total frequency.

Slides

Discrete & Continuous Data

Numerical data can be discrete or continuous.
Continuous data can take any value on a scale (e.g. height, weight).
Discrete data can only take particular values, often integers (e.g. number of pets).

Which of these is an example of continuous data?

Number of students in a class
Shoe size
Height of a plant
Number of cars in a car park

Mean, Median & Mode

Mode: the value that appears most often.
Median: the middle value when data is in order. For an even number of values, take the midpoint of the two middle values.
Mean: sum of values divided by number of values.

Find the median of 4, 2, 3, 1, 5.

Calculations with the Mean

Mean = total of values ÷ number of values.
Total of values = mean × number of values.
Use this to find missing values or when a new value is added.

A class of 24 students has a mean height of 1.56 m. A new student joins, and the mean becomes 1.57 m. The height of the new student $is ____ m$ .

Averages from Tables

Mode: data value with highest frequency.
Median: the (n+1)/2 th value, where n is total frequency.
Mean: sum of (data value $\times frequency)$ divided by total frequency.
Range: largest data value minus smallest data value.

Which of these are true about the frequency table below? Number of pets: 0,1,2,3,4 Frequency: 2,7,6,4,1

The mode is 1 pet.
The median is 2 pets.
The mean is 1.75 pets.
The range is 4 pets.

Averages from Grouped Data

For grouped data, we can only estimate the mean using midpoints.
Modal class: the class interval with the highest frequency.
Median lies in the class interval containing the (n+1)/2 th value.

For grouped data, why can we only estimate the mean?

Because the data is not numerical
Because we don't have the original data values
Because the mode is easier to find
Because the median is always given

Range & Interquartile Range

Range = highest value - lowest value. It measures spread but is affected by outliers.
Quartiles split data into four parts. Lower quartile (LQ) is median of lower half; upper quartile (UQ) is median of upper half.
Interquartile range $(IQR) = UQ - LQ$ . It measures spread of middle 50% and is not affected by outliers.

For the data set 31,32,35,35,36,37,39,40,42,45,46,48,49,50,51,51,53,54,57,60, the IQR $is ____$ .

Comparing Data Sets

Compare averages (mode, median, mean) and spreads (range, IQR).
Use the context to explain what the numbers mean in real life.
Avoid using the mean if there are extreme values.

Snails: median 7.1 cm, range 3.1 cm. Slugs: median 9.7 cm, range 4.5 cm. Which statement is correct?

Snails travel further on average and have more variation.
Slugs travel further on average and have more variation.
Snails travel further on average and have less variation.
Slugs travel further on average and have less variation.

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