**Mode****: ****The mode is the value, or class interval, which occurs most** **often**

**Mean **

The mean of the values with frequencies the x_{1 ,} x_{2}…… x_{n} with frequencies f_{1 },f_{2},… ,f_{n} the mean is m = x =\(\frac{1}{N} \) \(\sum_{i = 1}^n \) x_{i ,} f_{i}=1 where N= \(\sum_{i = 1}^n \) f_{i}

\(\sum{x_i f_i}\) = 142 and N = \(\sum{f_i}\) = 56

=> \(\bar{x}\) = \(\frac{142}{56}\) = 2.54

__Coding __

The weights of a group of people are given as x_{1 ,}x_{2……….} x_{n }in _{ , }*kilograms*. These weights are now changed to *grammes* and given as t_{1} , t_{2}……. t_{n} In this case– this is an example of *coding*.

**Coding and calculating the mean **

With the coding, t_{i} = \(\frac{x_{i-20 }{5}\), we are subtracting 20 from each *x*-value and then dividing the result by 5.We first find the mean for , and then we reverse the process to find the mean for ⇒ we find the mean for , multiply by 5 and add 20, giving

x ̅=5t ̅+20

__Median__**: **The median is the middle number in an ordered list.

__When to use mode, median and mean __

**Mode **

You should use the mode if the data is qualitative (colour etc.) or if quantitative (numbers) with a clearly defined mode (or bi-modal). It is not much use if the distribution is fairly even.

**Median **

You should use this for quantitative data (numbers), when the data is skewed, i.e. when the median, mean and mode are probably not equal, and when there might be extreme values

(outliers).

**Mean **

This is for quantitative data (numbers), and uses all pieces of data. It gives a true measure, and should only be used if the data is fairly symmetrical (not skewed), i.e. the mean could not be affected by extreme values (outliers).

__Median (____Q____2), quartiles (____Q____1, ____Q____3) and percentiles __

**Discrete lists and discrete frequency tables **

To find medians and quartiles

- Find k = \(\frac{n}{2}\) (for Q2 ) \(\frac{n}{4}\) (for Q1 ), \(\frac{3n}{4}\) ( for Q3 )

- If
*k*is an integer, use the mean of the k^{th}and ( k + 1 )^{th }numbers in the list. - If
*k*is not an integer, use the next integer**up**, and find the number with that position in the list.

**Interquartile range **

The interquartile range, I.Q.R., is *Q*3 – *Q*1.

**Range **

The range is the largest number minus the smallest (including outliers).

**Discrete lists **

A discrete list of 10 numbers is shown below:

*x ** *11 13 17 25 33 34 42 49 51 52

*n *= 10 for *Q*1, \(\frac{n}{4}\) = 2.5 so use 3rd number, ⇒ *Q***1 = 17 **

for *Q*2, 2 = 5 so use mean of 5th and 6th, ⇒ *Q***2 = 33 **median

for *Q*3 , 34 = 7.5 so use 8th number, ⇒ *Q***3 = 49 **

The interquartile range, I.Q.R., is , and the range is

__Grouped frequency tables, continuous and discrete data __

To find medians and quartiles

- Find k = \(\frac{n}{2}\) (for Q2 ) \(\frac{n}{4}\) (for Q1 ), \(\frac{3n}{4}\) (for Q3)
**Do not round***k***up or change it in any way.** - Use linear interpolation to find median and quartiles –
**note**that you must use the correct intervals for discrete data (start at the halves).

**Box Plots **In a group of people the youngest is 21 and the oldest is 52. The quartiles are 32 and 45, and the median age is 41. We can illustrate this information with a box plot as below – remember to include a scale and label the axis.

__Outliers __

An outlier is an extreme value. You are not required to remember how to find an outlier – you will always be given a rule.

For example : Outliers are values outside the range

Q1 – 1⋅5 × (Q3 – Q1) to Q3 + 1⋅5 × (Q3 – Q1).

**Skewness **

A distribution which is symmetrical is **not **skewed

**Positive skew **

If a symmetrical box plot is stretched in the direction of the positive *x*-axis, then the resulting distribution has *positive *skew.

For positive skew : Q3 – Q2 > Q2 – Q1

The same ideas apply for a continuous distribution, and a little bit of thought should show that for *positive *skew *mean *> *median *> *mode*.

**Negative skew **If a symmetrical box plot is stretched in the direction of the negative *x*-axis, then the resulting distribution has *negative *skew.

For negative : Q3 – Q2 < Q2 – Q1

The same ideas apply for a continuous distribution, and a little bit of thought should show that for *negative *skew *mean *< *median *< *mode*.