Question

1) Using the excel data file “US violent crime” which shows the violent crime rate in the US from 1960 to 2012:

(20 pts) Make a time series plot of the data

(5 pts each 25 pts total) Determine the following: Mean, Median, Standard deviation, Q1 and Q3. (25 pts)

Make a histogram of the data. Hint the year is not used, you need to determine how many years fall into each of the classes.

(7) What are your thoughts on the time series plot, i.e. trends etc.?

(8) Thoughts on the histogram i.e. shape of distribution etc.?

[Excel sheet]

Year	Violent Crime rate
1960	160.9
1961	158.1
1962	162.3
1963	168.2
1964	190.6
1965	200.2
1966	220.0
1967	253.2
1968	298.4
1969	328.7
1970	363.5
1971	396.0
1972	401.0
1973	417.4
1974	461.1
1975	487.8
1976	467.8
1977	475.9
1978	497.8
1979	548.9
1980	596.6
1981	593.5
1982	570.8
1983	538.1
1984	539.9
1985	558.1
1986	620.1
1987	612.5
1988	640.6
1989	666.9
1990	729.6
1991	758.2
1992	757.7
1993	747.1
1994	713.6
1995	684.5
1996	636.6
1997	611.0
1998	567.6
1999	523.0
2000	506.5
2001	504.5
2002	494.4
2003	475.8
2004	463.2
2005	469.0
2006	479.3
2007	471.8
2008	458.6
2009	431.9
2010	404.5
2011	387.1
2012	386.9

Answer 1

We are provided with the data on “US Violent Crimes” corresponding to the years from 1960 to 2012

Thus we obtain the Time series plot as follows:

Also we calculate the mean using the following formula:

Mean= Sum total of all values of "US Violent Crimes"Total Number Of Observations = 476.55

Now we order the observations in ascending order of magnitude.

Let N be the total number of observations in the sample. Thus, N=53.

Note that here N is an odd integer.

Hence we can calculate the median as the [(N+1)/2]^th observation in the ordered arrangement.

Similarly, Q1 and Q3 are calculated as the [(N+1)/4]^th and [3(N+1)/4]^th observation respectively in the ordered arrangement.

Thus median= 479.3; Q1= 391.55; Q3= 595.05

Let x_t denote the number of “US Violent Crimes” in the year t.

Thus the standard deviation is given by:

SD= 1Nt=19602012(Xt-X)^2

We obtain SD as: SD=163.23

Now we have range=Max value of x_t – Min value of x_t

                                       = 600.1

We divide the range into 7 classes each of size 100. Thus the class limits are taken as: 150-249.9, 250-349.9 ,…, 750-849.9

Thus we obtain the following histogram:

• From the time series plot, we observe that till the year 1995 there was an overall increase in the number of violent crimes with some cyclical pattern, while it fell drastically after 1995 , since then it has been decreasing slowly over the years.
• The histogram is quite bell shaped with broad left tail and it is quite symmetric. So Cauchy distribution will be able to explain the data than any other standard distributions.