Working with one variable data

Spread Joaquin’s Tests Taran’s Tests: 76, 45, 83, 68, 64 67, 70, 70, 62, 62 What can you infer, justify and conclude about the Joaquin’s and Taran’s tests scores? (Hint: Calculate the mean, median and mode for each. What do they tell you?) J.’s mean = T.’s mean = med = med = mode = none mode =

Spread Mean, median and mode are all good ways to find the centre of your data. This information is most useful when the sets of data being compared are similar. It is also important to find out how much your data is spread out. This gives a lot more insight to data sets that vary from each other.

Consider the following two data sets with identical mean and median values. Why is this information misleading? ( Mean = 5, Median = 5) Set A) 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9 Set B) 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7 This information is misleading because one graph is bell-shaped and the other is uniform, but the calculations make them appear to be similar when really A and B are spread out quite differently.

Measures of Spread In analysing data, it is often important to know whether it is spread out, or whether it is clustered around the mean. Measures of spread are used to quantify the spread of the data. The measures of spread, or dispersion are: Range Quartiles Variance Standard deviation

Range The simplest measure of dispersion. Calculated by finding the difference between the greatest and the least values of the data. Useful since it is the easiest to understand. Affected by extreme data. The range of values 1, 2, 4, 6, 9, 11, 15, 25 is 25 – 1 = 24

Quartiles and Interquartile Ranges Quartiles divide a set of ordered data into four groups with equal numbers of values. Lowest Datum First Quartile Q 1 Median Q 2 Third Quartile Q 3 Highest Datum The three “dividing points” are the first quartile (Q 1 ), median, (sometimes called the second quartile, Q 2 ), and the third quartile (Q 3 )

Quartiles and Interquartile Ranges Lowest DatumQ1Q1 Median Q 2 Q3Q3 Highest Datum  The interquartile range is Q 1 – Q 3, which is the range of the middle of the data.  The semi-interquartile range is one half of the interquartile range.  Both these ranges indicate how closely the data are clustered around the median.

Box and Whisker Plot Illustrates the Quartiles The Box shows the interquartile range The whiskers represent the lowest and highest values A modified box and whisker plot shows outliers outside of the whiskers See Page 141 for illustrations

Standard Deviation A deviation is the difference between an individual value in a set of data and the mean for the data. Standard Deviation averages the square of the distance that each piece of data is from the mean. The smaller the standard deviation, the more compact the data set.

Standard Deviation – Population σ = Standard Deviation - Population ∑ = Sum μ = Mean N = Number of data in population

Standard Deviation – Sample s = Standard Deviation - Sample ∑ = Sum = Mean n = Number of data in sample

Variance The variance can be found by calculating the average squared difference ( or deviation ) of each value from the mean. PopulationSample Or square the standard deviation.

Standard Deviation – Group Data If you are working with grouped data, you can estimate the standard deviation using the following formula PopulationSample f i = the frequency for a given interval m i = the midpoint of the interval

Find the Measures of Spread Rachelle works part-time at a gas station. Her gross earnings for the past eight weeks are shown. $55$68$83$59$68$95$75$65 Calculate the range, variance, standard deviation, interquartile, and semi-interquartile ranges for her weekly earnings.

Find the Measures of Dispersion Range: The range of Rachelle’s earnings is $

Find the Measures of Dispersion Variance: Gross Earnings Total The variance of Rachelle’s earnings is $

Find the Measures of Dispersion Standard Deviation: The standard deviation of Rachelle’s earnings is $

Find the Measures of Spread Interquartile range: First, put the data into numerical order Interquartile range = Q 3 - Q 1 = 79 – 62 = 17

Find the Measures of Spread Semi-Interquartile range: Semi-Interquartile range = 17/2 = 8.5 Therefore the interquartile range is 17 and semi-interquartile range is 8.5.

Standard Deviation Group Data - Example The following table represents the number of hours per day of watching TV in a sample of 500 people. Number of hours Frequency

Interval Midpoint (m i ) Frequency f i ( )2 = x = x 6.76 = x 0.36 = x 1.96 = x = x = x = = 3.05 THEREFORE THE STANDARD DEVIATION IS APPROXIMATLY 3.05

Z-Scores The number of standard deviations away from the mean a data point is –Thus if our standard deviation is 8 then how many 8’s is a data point (13) away from the average or centre –It is found by dividing the deviation by the standard deviation If your values are below the mean their z score will be negative. Similarly if your value is above the mean your z score will be positive

Percentiles Similar to quartiles Percentiles divide the data into 100 intervals that have equal number of values. k percent of the data are less than or equal to k th percentile P k Which means that you are finding what percent of the data is below your specific value in question Often used for Standardized Tests

Homework Pg 148 #1-6, 14 I LOVE HOMEWORK