IQ
The results are now able to be recorded in a diagram. To begin with analysing the data about IQ.
More gathering of information is needed to support these statements:
The mode IQ for boys in my sample was higher than the mode IQ for girls. Also, the sample suggests that the boys have a higher IQ between 110 and 120 than the girls.
Comparing the mean, median, mode and range of IQ for boys and girls will give more evidence.
Mean IQ
The mean can be calculated from the frequency tables.
f is the frequency and x is the IQ then the mean is:
Boys
Σfx (3x85)+(4x95)+(7x105)+(14x115)+(1x125)+(1x135) 3240
Σf 3+4+7+14+1+1 30
Σfx (1x85)+(13x95)+(9x105)+(5x115)+(1x125)+(1x135) 3100
Σf 1+13+9+5+1+1 30
Mean IQ for boys = 108
Mean IQ for girls = 103.33
Modal IQ
The modes of the IQ for boys and girls can be read off the bar charts or frequency tables:
Modal IQ for boys = 115
Modal IQ for girls = 95
Median IQ
There are 30 people in each sample, so the median will be half way between the 15th and 16th values:
Median IQ for boys = 109+113 111
2
Median IQ for girls = 96+104 100
2
Range IQ
The range of the IQ will show how spread out the data is:
Range of the IQ for boys = 50
Range of the IQ for girls = 40
The Summary of the results:
There is more evidence to describe the difference in IQ between boys and girls:
All three measures of average (mean, median and mode) are greater for boys than girls. The range of the IQ for boys is greater than the girls. In conclusion, even though there is a small number of boys with lower IQ’s and girls with higher IQ’s the evidence in general is that the boys IQ is greater than the girls. The difference in range is important, because it means that the boys IQ is more variable.
Height
The results are now able to be recorded in a diagram. To begin with analysing the data about height, using the bar charts to compare the result for boys and girls. Due to height being continuous it has to be recorded on a histogram.
Boys
Girls
There is evidence to prove the difference in height between boys and girls:
The boys are taller than the girls between 1.7 and 1.8 metres. As, there is not that many girls of that height. Also, there seems to be more boys taller between 1.9 and 2 metres than the girls. However the girls are taller than the boys between 1.6 and 1.7 metres. In conclusion, although there are a small number of boys with smaller heights and girls with bigger heights, the evidence suggests that, in general, the heights for the boys are greater that the heights for the girls.
The continuous data is compared by drawing a frequency polygon.
Boys
Girls
This is the sample of 30 boys and 30 girls.
Frequency Polygon
This is a good way of comparing continuous data by drawing a frequency polygon.
Boys
Girls
The reason there is an extra column showing the Cumulative Frequency, is due to the following Cumulative Frequency Curve graphs on the following pages.
Since the data is grouped into class intervals, it also makes sense to record it in a stem and leaf diagram. This will make it easier to read off the median values.
Boys
Girls
Averages
It is also possible to record the mean, median and range for the data. Due to the data being continuous it makes more sense to find the modal class interval rather than the mode. This is the class interval that contains that most values. The values for the mode and median have been rounded to two decimal places.
Comments about the heights between the male and female students:
All three measures of average in the sample were higher for the boys than for the girls, though the range for the boys and girls was the same at 0.70m. The sample suggests that 8 out of 30, or 27% of the boys have a height between 170 and 180cm, whilst 12 out of 30, or 40% of the girls have a height between 160 and 170cm. The frequency polygon shows that there are fewer girls with heights below 140cm than the boys.
These conclusions are based on a sample of only 30 girls and 30 boys. I could extend the sample or repeat the whole exercise to confirm my results. But, there is no need to repeat the test or use a larger sample, but there is more evidence to support my statements.
Cumulative Frequency Curves
For the Frequency Tables, please look at the heading; Tally Chart
The benefit of drawing cumulative frequency curves for a continuous variable like height and IQ is that it is easy to read off the graph to retrieve the median, upper quartile, lower quartile and interquartile range:
Height
IQ
The boys median height, is higher than the girls and the boys median IQ, is higher than the girls. Also, the boys interquartile range for height is a lot higher than the girls, but for the IQ they are more or less the same. Even though the girls height is lower than the boys height it more consistent. For, the IQ both samples are equally consistent. In conclusion, the boys height and IQ is a lot higher than the girls. So, in general he boys are taller and have a higher IQ than the girls.
Comparing Height and IQ
When the investigation was extended the hypothesis was:
The taller the person is, the higher the IQ.
A random sample of 30 students is needed to test the hypothesis.
The best way to compare the data is by using a scatter diagram.
There is moderate positive correlation between height and IQ. This suggests that the taller the person is the higher the IQ.
Boys have a larger number of siblings than pets.
A sample of 30 boy students is shown below. They have been renumbered 1 to 30 to make the data easier to represent and analyse.
This dual bar chart shows that there are more siblings in a family than there are of pets. Though it is not a very useful graph, but it does show that the siblings are greater than the pets.
Tally Chart
For a more useful representation of this data. Here are the frequency tables for Number of siblings and Number of pets.
Number of Siblings
This bar chart shows that the boys have quite a few siblings in the 3rd region, meaning that they have 3 brothers or sisters. All of them have more than one sibling.
From the frequency table above the mean number of siblings is:
Σfx (1x1)+(12x2)+(10x3)+(2x4)+(1x5)+(2x6)+(2x7) 94
Σf 1+12+10+2+1+2+2 30
Mean Number of Siblings = 3.13
Number of Pets
This bar chart shows that the boys have quite a few pets in the 2nd region, meaning that they have 2 pets. All of them have 1 or more pets.
From the frequency table above the mean number of siblings is:
Σfx (1x0)+(18x1)+(8x2)+(1x3)+(1x4)+(1x5) 46
Σf 1+18+8+1+1+1 30
Mean Number of Pets = 1.53
In conclusion, although there are a small number of boys with a small number of siblings and boys with a small number of pets, the evidence suggests that, in general, the number of siblings for the boys is greater the number of pets for boys.
Spearman’s Rank Correlation Coefficient
This will be used to find the extent to which two sets of data correlate.
Σd² = 116
There are 30 samples, so n = 30
rs = 1 - 6Σd²
n(n²-1)
= 1 - 6x-976
30(900-1)
= 1 - -976
26970
= 1 – 0.0258 = 0.9
0.9 is a high agreement between the number of pets and number of siblings.
Cumulative Frequency Curves
For the Frequency Tables, please look at the heading; Tally Chart
The benefit of drawing cumulative frequency curves for a continuous variable like number of siblings and number of pets is that it is easy to read off the graph to retrieve the median, upper quartile, lower quartile and interquartile range:
Number of Siblings
Number of Pets
