Averages, Ranges & Data Flashcards

All 36 cards in this deck

What is the mode of a set of data?

The value that occurs most often.

What is the formula for the mean of a list of data?

$\text{mean} = \dfrac{\text{sum of the values}}{\text{number of values}}$

True or false? If a data set has an even number of values, the median must be one of the values in the data set.

False. The median is the mean of the two middle values, which may not be a value in the set.

Which average is usually the most appropriate for a data set containing a few extreme values, and why?

The median, because the mean is affected (distorted) by the extreme values.

Which average can be used for non-numerical (categorical) data such as favourite colour?

The mode (the modal category).

What are the steps to find the mean of the remaining values when two means are known?

e.g. the mean of 8 parcels is $2.5$ kg; the mean of 3 of them is $2$ kg — find the mean of the other 5

1. Total of all: $8 \times 2.5 = 20$
2. Total of the known group: $3 \times 2 = 6$
3. $(20 - 6) \div 5 = 2.8$ kg

What are the steps to find the mean of two groups combined?

e.g. 18 boys with mean age $16.2$ years and 27 girls with mean age $16.7$ years

1. Total for each group: $18 \times 16.2 = 291.6$ and $27 \times 16.7 = 450.9$
2. Add the totals: $742.5$
3. Divide by the total frequency: $742.5 \div 45 = 16.5$ years

True or false? The mean of two groups combined is the mean of the two group means.

False. You must use the group totals; averaging the two means only works if the groups are the same size.

True or false? If one value in a data set is replaced by a larger value, the mean increases.

True. The total increases while the number of values stays the same.

What are the steps to find the mean from an ungrouped frequency table?

e.g. value 1 (frequency 4), value 2 (frequency 3), value 3 (frequency 3)

1. Work out $fx$ for each row: $4$, $6$, $9$
2. Find $\sum fx = 19$ and $\sum f = 10$
3. Mean $= 19 \div 10 = 1.9$

True or false? The mode of an ungrouped frequency table is the largest frequency in the table.

False. The mode is the data value that has that largest frequency.

True or false? To find the mean from a frequency table you divide $\sum fx$ by the number of rows in the table.

False. You divide by the total frequency $\sum f$.

What are the steps to work out an estimate for the mean from a grouped frequency table?

e.g. $0 < t \le 10$ (frequency 3), $10 < t \le 20$ (frequency 2)

1. Find the midpoint of each class: $5$ and $15$
2. $\sum fx = 5 \times 3 + 15 \times 2 = 45$
3. Divide by $\sum f = 5$: estimated mean $= 9$

How do you find the midpoint of a class interval?

e.g. $20 < t \le 30$

Add the two end values and divide by 2.

$(20 + 30) \div 2 = 25$

What is the modal class of a grouped frequency table?

The class interval with the highest frequency.

True or false? The median of grouped data is the midpoint of the middle class in the table.

False. You must use the frequencies to find which class contains the middle value.

Why is the mean found from a grouped frequency table only an estimate?

The exact data values are not known, so each value is assumed to be the midpoint of its class.

What is the formula for the range of a set of data?

Range $=$ largest value $-$ smallest value.

What is the formula for the interquartile range?

IQR $=$ upper quartile $-$ lower quartile.

Where are the lower and upper quartiles in an ordered list of $n$ values?

Lower quartile at position $\frac{n+1}{4}$, upper quartile at position $\frac{3(n+1)}{4}$.

e.g. $n = 23$ → 6th and 18th values

True or false? Three quarters of the data lies between the lower quartile and the upper quartile.

False. The middle 50% (half) of the data lies between them.

What is an outlier?

A value that is much larger or much smaller than the rest of the data.

What effect does one very large outlier have on the mean, the median and the range?

It increases the mean and the range a lot, but has little or no effect on the median.

When comparing two distributions, what two types of measure should your comparison include?

An average (mean, median or mode) and a measure of spread (range or IQR), both written in context.

When comparing two data sets, what does a smaller interquartile range tell you?

The middle 50% of the values are less spread out, so that data set is more consistent.

When comparing two data sets, what does a higher median tell you?

On average the values in that set are larger.

e.g. "the boys' times were generally longer"

True or false? A data set with a larger range is more consistent.

False. A larger range means the data is more spread out, so less consistent.

Give a reason for choosing the mean rather than the median as the average of a data set.

The mean uses all of the data values.

What is a population in statistics?

The whole group that is being investigated.

What is a sample?

A smaller part of the population that is selected and studied.

What is a random sample?

A sample in which every member of the population has an equal chance of being chosen.

Give one limitation of using a sample to describe a population.

The sample may not be representative (it may be biased), so results are only estimates.

True or false? A small sample is just as reliable as a large sample for describing a population.

False. A larger sample is more likely to represent the population.

What are the steps to estimate a number in a population from a sample?

e.g. 12 of a sample of 50 people are left-handed; the population is 800

1. Find the sample proportion: $\frac{12}{50}$
2. Multiply by the population size: $\frac{12}{50} \times 800$
3. Estimate $= 192$ people

What are the steps to find the median from an ungrouped frequency table?

e.g. value 1 (frequency 2), value 2 (frequency 3), value 3 (frequency 4)

1. Add the frequencies: $2+3+4=9$
2. Median position $=\frac{n+1}{2}=\frac{10}{2}=5$th value
3. Count through the frequencies (cumulative $2, 5, 9$): the 5th value is $2$, so the median is $2$

True or false? The median of an ungrouped frequency table is the value in the middle row of the table.

False. You must use the total frequency: find the $\frac{n+1}{2}$th value by counting through the frequencies.