Averages and Range

Range Range = highest amount - lowest amount

Types of average –Sometimes we are told that the average family has 2.4 children. –There are three types of average: mode, median and mean

Mean If amounts are shared out equally, then each person (or item) will get the same amount which is the mean. If the total number of children are shared out between all the families then there are 2.4 children per family. Mode The mode is the most common amount. Most families have 2 children so the mode of the number of children is 2. Median When the amounts are arranged in order, the middle one is the median.

–A die is tossed 15 times and the scores were: –6, 5, 4, 3, 4, 1, 1, 2, 4, 4, 3, 2, 3, 5, 4 –Score Frequency –12 –22 –33 –45 –52 –61 From the table we can see that the score with the highest frequency is 4, so the mode is 4.

–Writing the scores in order gives –1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6 –The median is the middle one i.e. 4 –The mean is found by adding together all the scores and dividing by the number of scores (15) – = 51

The mean height of this mountain range is 1000m 1000 m

The mean height of this mountain range is also 1000m. The mean can mask extreme values. 1000m

Mean, Mode and Median of Frequency Distributions –The ages of 50 people in a maths club are given below –Find the mean, mode and median of their ages. –23 has the highest frequency, so the mode is 23 –where  f is the total of the frequencies and  fx is the total of the frequencies × values Age Frequency

–To find the mean, rewrite the frequency table as shown Age (x) Frequency (f) Frequency × age (f × x)

–To find the mean and median, put the data into a frequency table. Number of children (x) Number of families (f) f × x The first 5 families have 0 children, the next 15 families have 1 child. Therefore the 18 th family has 1 child.

Grouped Data –For a grouped frequency distribution the modal class is the group (or class) with the highest frequency. –An estimate for the mean can be found by assuming that each value in a class is equal to the value of the mid-point of the class and then using the formula

Example –At a supermarket, the manager recorded the lengths of time that 80 customers had to wait in the check-out queue. –Find the modal class and calculate an estimate for the mean. Waiting time (t)Mid-point x Frequency ff × x 0 ≤ t < ≤ t < ≤ t < ≤ t < ≤ t < ≤ t <

–The class with the highest frequency is the 200 ≤ t < 250 class. The modal class is 200 ≤ t < 250 –The mean is Waiting time (t)Mid-point x Frequency ff × x 0 ≤ t < ≤ t < ≤ t < ≤ t < ≤ t < ≤ t <

Histograms Histograms are used to represent information contained in grouped frequency distributions. The horizontal axis is a continuous scale There are no gaps between the bars

Histogram yn dangos Taldra Bechgyn Histogram representing Boys’ Heights

Histogram yn dangos Taldra Genethod Histogram representing Girls’ Heights

Frequency Polygons Frequency polygons are often used instead of histograms when we need to compare two or more groups of data To draw a frequency polygon:  Plot the frequencies at the midpoint of each class interval  Join successive points with straight lines To compare data, frequency polygons for different groups of data can be drawn on the same diagram

Polygon Amlder a Histogram o Daldra Bechgyn Frequency Polygon and Histogram of Boys Heights

Polygon Amledd a Histogram o Daldra Genethod Frequency Polygon and Histogram of Girls Heights

Defnyddio polygonau amledd i gymharu dwy set o ddata Using frequency polygons to compare two sets of data