Chi Squared

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Determining whether observed frequencies differs significantly from an even distribution or any other distribution hypothesized

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Chi Squared

  1. 1. ONE-WAY CHI TEST<br />Determining whether observed frequencies differs significantly from an even distribution or any other distribution hypothesized<br />Co-Authors:<br />Rosetta Jones and Sharon Tutt<br />Dated: October 24, 1996<br />
  2. 2. INTRODUCTION<br />The act of domestic violence has no label. It could happen to any gender, at any age and in any occupation. Domestic violence can also take on the form of mental, physical, verbal, emotional, and psychological abuse. The ultimate goal of a domestic violence abuser is to have total control and power over another person. This situation can be inflicted upon anyone at anytime. Russell Hovan’s perspective states, “There are situations in life to which the only satisfactory response is a physically violent one. If you don’t make that response, you continually relive the unresolved situation over and over in your life.” (b. 1925, U.S. author. Novelists in Interview (ed. by John Haffenden, 1985)).<br />The scariest account of domestic violence occurs when you do not have to see a scar to feel one. It’s like a thief in the night, because we don’t think about it until it indirectly or directly affects us on a personal level. There are many cases that have unknown causes and probably as many that go unreported. It’s a time bomb waiting to explode, and we will show the impact of the explosion by reviewing 1994 and 1995 domestic violence data. Our source was provided through the publication, The Statistical Crime Data for the State of Maryland. By using the chi-squared test, it will demonstrate if there is a significant or insignificant trend.<br />
  3. 3. Null Hypothesis: The use of knives in domestic violence incidents has significantly changed during 1994 and 1995.<br />Research Hypothesis:The use of knives in domestic violence incidents has not changed during 1994 and 1995.<br />Our hypothesis is drawn from a sampling that observed 1994 and 1995 aggravated and non-aggravated assault charges drawn from the Statistical Crime Data for the State of Maryland, c. 1995. The one-way chi-test can be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized). <br />What should the frequency distribution of correct observances look like if the null hypothesis were true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 20.378 incidents for 1994 and 24.021 for 1995, we should expect 4.08 occurrences (in 1994) and 4.80 occurrences (in 1995) in each Type of Weapon Category. The expected frequencies are those frequencies that are expected to occur under the term of the Null Hypothesis. <br />
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  5. 5. Null Hypothesis: The use of knives in domestic violence incidents has significantly changed during 1994 and 1995.<br />Research Hypothesis:The use of knives in domestic violence incidents has not changed during 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 5 categories (firearms; knives; other dangerous weapons; hands, fists, feet, etc.; and non-aggravated) in 1994 and 1995: <br />df = 5 - 1 = 4 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 4 degrees of freedom. The value is 9.488. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is 52.6 and for 1995 is 59.2, it is larger than the table value, we reject the null hypothesis and accept the research hypothesis. The observed frequencies do differ significantly from the expected occurrence for an equal distribution under the null hypothesis for knife usage in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did not indicate a perfectly even distribution (4.08 in 1994 and 4.80 in 1995 for each type of weapon), the degree of unevenness was sufficiently large. <br />
  6. 6. Null Hypothesis: Unknown factors do not play a significant role in domestic violence occurrences during 1994 and 1995.<br />Research Hypothesis:Unknown factors do play a significant role in domestic violence occurrences during 1994 and 1995.<br />This hypothesis was also taken from a sampling that observed 1994 and 1995 unknown factors in domestic violence drawn from The Statistical Crime Data for the State of Maryland, c. 1995. Again, the one-way chi-test will be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized). <br />We want to demonstrate if this frequency distribution for unknown factors shows if the null hypothesis is true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 20.378 incidents for 1994 and 24.021 incidents for 1995, we should expect .926 occurrences (in 1994) and 1.091 occurrences (in 1995) for each year’s Circumstances Category. The expected frequencies are those incidents that are expected to occur under the term of the Null Hypothesis. <br />
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  9. 9. Null Hypothesis: Unknown factors do not play a significant role in domestic violence occurrences during 1994 and 1995.<br />Research Hypothesis:Unknown factors do play a significant role in domestic violence occurrences during 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 22 categories (Alcohol, Drugs, Food/Cooking, Friends, Gambling, Household Chores, Infidelity, Employment/Job, Mental Imbalance, Money, Children, Property, Relative, Sex, Hobby, T.V., Separation, Divorce, Reconciliation, Out Late, Other and Unknown) in 1994 and 1995: <br />df = 22 - 1 = 21 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 21 degrees of freedom. The value is 32.671. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is 78.5 and for 1995 is 101.29, it is larger than the table value, we reject the null hypothesis and accept the research hypothesis. The observed frequencies do differ significantly from the expected occurrence for an equal distribution under the null hypothesis for unknown factors in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did not indicate a perfectly even distribution (.926 in 1994 and 1.091 in 1995 for each circumstance), the degree of unevenness was sufficiently large. <br />
  10. 10. Null Hypothesis: The morning hours have an effect on domestic violence during 1994 and 1995.<br />Research Hypothesis:The morning hours have no effect on domestic violence during 1994 and 1995.<br />The above hypothesis was also taken from a sampling that observed domestic violence’s 1994 and 1995 incidents occurring in the morning hours. This sample was also drawn from The Statistical Crime Data for the State of Maryland, c. 1995. Again, the one-way chi-test will be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized). <br />We want to demonstrate if this frequency distribution for the morning hours shows if the null hypothesis is true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 76.95 incidents for 1994 and 93.57 incidents for 1995, we should expect 6.41 morning occurrences (in 1994) and 7.80 morning occurrences (in 1995) for each year’s Morning Hours Category. The expected frequencies are those incidents that are expected to occur under the term of the Null Hypothesis. <br />
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  12. 12. Null Hypothesis: The morning hours have an effect on domestic violence occurrences during 1994 and 1995.<br />Research Hypothesis:The morning hours have no effect on domestic violence occurrences during 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 12 categories (from 12:00 a.m. until 11:00 a.m.) in 1994 and 1995: <br />df = 12 - 1 = 11 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 11 degrees of freedom. The value is 19.675. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is 14.7 and for 1995 is 22.3, it is smaller than the table value, we accept the null hypothesis and reject the research hypothesis. The observed frequencies do not differ significantly from the expected occurrence for an equal distribution under the null hypothesis for morning incidents in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did indicate a perfectly even distribution (6.41 in 1994 and 7.8 in 1995 for each circumstance) for the Morning Hours Category in 1994. However, comparing the chi-squared test to our obtained frequencies for the Morning Hours Category in 1995, the reported sampling did not indicate a perfectly even distribution (6.41 in 1994 and 7.8 in 1995 for each circumstance) the degree of unevenness was significantly large.<br />
  13. 13. Null Hypothesis: The afternoon hours have an effect on domestic violence occurrences during 1994 and 1995.<br />Research Hypothesis:The afternoon hours have no affect on domestic violence occurrences during 1994 and 1995.<br />The above hypothesis was also taken from a sampling that observed domestic violence 1994 and 1995 incidents occurring in the afternoon hours. This sample was also drawn from The Statistical Crime Data for the State of Maryland, c. 1995. Again, the one-way chi-test will be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized). <br />We want to demonstrate if this frequency distribution for the afternoon hours shows if the null hypothesis is true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 76.95 incidents for 1994 and 93.57 incidents for 1995, we should expect 5 morning occurrences (in 1994) and 6 morning occurrences (in 1995) for each year’s Afternoon Hours Category. The expected frequencies are those incidents that are expected to occur under the term of the Null Hypothesis. <br />
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  15. 15. Null Hypothesis: The afternoon hours have an effect on domestic violence occurrences during 1994 and 1995.<br />Research Hypothesis:The afternoon hours have no effect on domestic violence occurrences during 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 12 categories (from 12:00 p.m. through 11:00 p.m.) in 1994 and 1995: <br />df = 12 - 1 = 11 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 11 degrees of freedom. The value is 19.675. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is 5 and for 1995 is 6, it is smaller than the table value, we accept the null hypothesis and reject the research hypothesis. The observed frequencies do not differ significantly from the expected occurrence for an equal distribution under the null hypothesis for morning incidents in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did indicate a perfectly even distribution (10.57 in 1994 and 12.22 in 1995 for each circumstance). <br />
  16. 16. Null Hypothesis: The months of the year have an effect on domestic violence during 1994 and 1995.<br />Research Hypothesis:The months of the year have no affect on domestic violence during 1994 and 1995.<br />The above hypothesis was also taken from a sampling that observed domestic violence 1994 and 1995 incidents occurring in the month of the year. This sample was also drawn from The Statistical Crime Data for the State of Maryland, c. 1995. Again, the one-way chi-test will be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized).<br />We want to demonstrate if this frequency distribution for the month of the year shows if the null hypothesis is true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 20.378 incidents for 1994 and 24.021 incidents for 1995, we should expect 1.7 monthly occurrences (in 1994) and 2 monthly occurrences (in 1995) for each of the year’s Month of the Year Category. The expected frequencies are those incidents that are expected to occur under the term of the Null Hypothesis.<br />
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  18. 18. Null Hypothesis: The months of the year have an effect on domestic violence during 1994 and 1995.<br />Research Hypothesis:The months of the year have no effect on domestic violence during 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 12 categories (from January through December) in 1994 and 1995: <br />df = 12 - 1 = 11 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 11 degrees of freedom. The value is 19.675. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is .041 and for 1995 is 1.365, it is smaller than the table value, we accept the null hypothesis and reject the research hypothesis. The observed frequencies do not differ significantly from the expected occurrence for an equal distribution under the null hypothesis for monthly incidents in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did indicate a perfectly even distribution (1.7 in 1994 and 2 in 1995 for each circumstance). <br />
  19. 19. Null Hypothesis: The day of the week has an affect on domestic violence occurring in 1994 and 1995.<br />Research Hypothesis:The day of the week has no effect on domestic violence occurring in 1994 and 1995.<br />The above hypothesis was also taken from a sampling that observed domestic violence 1994 and 1995 incidents occurring on a certain day of the week. This sample was also drawn from The Statistical Crime Data for the State of Maryland, c. 1995. Again, the one-way chi-test will be used to determine whether the frequencies observed differs significantly from an even distribution (or any other distribution hypothesized).<br />We want to demonstrate if this frequency distribution for the day of the week shows if the null hypothesis is true? The charts below demonstrate two categories with the anticipation that 50% of the reported results should fall in each classification. With 20.378 incidents for 1994 and 24.021 incidents for 1995, we should expect 2.911 monthly occurrences (in 1994) and 3.432 monthly occurrences (in 1995) for each of the year’s Day of the Week Category. The expected frequencies are those incidents that are expected to occur under the term of the Null Hypothesis.<br />
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  21. 21. Null Hypothesis: The day of the week has an effect on domestic violence occurring in 1994 and 1995.<br />Research Hypothesis:The day of the week has no effect on domestic violence occurring in 1994 and 1995.<br />The chi-squared method allows us to test the significance of the difference between a set of observed frequencies and expected frequencies. <br />Based on just the observed and expected frequencies, the formula for chi-square is: <br />(fo - fe)2<br />X2 = Sfe<br />where fo= observed frequency in any category <br />fe= expected frequency in any category <br />Our next step is to interpret the chi-square value by determining the appropriate degrees of freedom. <br />df = k - 1 <br />where k = number of categories in the observed frequency distribution. Our example shows 7 categories (from Monday through Sunday) in 1994 and 1995: <br />df = 7 - 1 = 6 <br />Now by checking The Critical Value of Chi-Square at the .05 and .01 Chart, we look for the .05 significant level to check 6 degrees of freedom. The value is 12.592. This is the value that we must exceed before we can reject the null hypothesis. Because our calculated chi-square for 1994 is .359 and for 1995 is 3.74, it is smaller than the table value, we accept the null hypothesis and reject the research hypothesis. The observed frequencies do not differ significantly from the expected occurrence for an equal distribution under the null hypothesis for day of the week incidents in domestic violence situations. <br />In comparing the chi-squared test to our obtained frequencies, the reported sampling did indicate a perfectly even distribution (2.911 in 1994 and 3.432 in 1995 for each circumstance). <br />
  22. 22. CONCLUSION<br />As the data shows, domestic violence in the State of Maryland for 1994 and 1995 indicates it is on the rise. We must all take steps to neutralize this time bomb. Because everybody is a victim as long as the abuse continues. In addition, everyone must begin to take an active and responsible role to curve this silent offense. As long as we continue to allow these abuses to perpetuate, we are threatening our own futures. If the trend continues to rise as it has been reported in 1994 and 1995, our children’s behaviors may grow more vicious in nature. <br />

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