Conditional Relative Frequency
Conditional Relative Frequency is a statistical concept that helps to interpret data by focusing on specific subsets of a population rather than the entire dataset. It is derived from the broader idea of relative frequency, which expresses how often an event occurs in relation to the total count of events. Conditional relative frequency allows researchers to examine the frequency of responses based on a particular condition, such as gender or the level of perceived safety in a given context.
For instance, if a survey asks respondents how safe they feel walking alone at night, one could calculate the conditional relative frequency for males and females separately to gain insights into their differing perceptions of safety. This method emphasizes the importance of context by conditioning the frequency calculation on a specified group or response. By focusing on these subgroups, analysts can uncover more nuanced patterns within the data, offering a clearer understanding of the relationships between variables. Ultimately, the choice of which conditional relative frequency to calculate depends on the specific questions posed in the analysis, making it a versatile tool in data interpretation.
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Conditional Relative Frequency
Frequency is a name given to the count of the number of times an event occurs. Relative frequency is the term used to indicate that the number has been transformed into a proportion expressing it in relation to the grand total of the number of events. The proportion may be expressed as a percentage. The conditional relative frequency rather than the grand total is sometimes used when constructing the proportion in order to make more sense of the data.
In the example (Figure 1), some data have been taken from the Crime Survey for England Teaching Dataset. For simplification, the data have been rounded. The rounded data suggest that we have obtained survey responses from 45,700 people. The table below gives the frequencies for the responses to the question "How safe do you feel walking alone after dark"? The table is split into two rows denoting the responses given by males and females.
Frequency is the number of times a specific response was observed. For example, frequency of responses that stated "very safe" regardless of gender was 15,600. Dividing this number by the overall number of respondents gives the relative frequency of a response stating "very safe" in reply to that question. For these data, the relative frequency is
In other words approximately one third of respondents stated they felt "very safe" walking alone after dark. We can also see that the relative frequency for the survey being completed by a male was
in other words, somewhat less than half the respondents were male.
Conditional relative frequency means that the statistician "conditions" on an event other than the event someone filled in on a form. Here, it is possible to either condition on the gender of the respondent, or the level of safety they stated they felt when walking alone. Essentially, relative frequencies are calculated using an appropriate total other than the grand total.
If using the gender of the respondent as the condition, only the row of data that applies to that gender is considered. If the condition on the event is "survey completed by male" there are 20,900 respondents. Of these, 10,100 stated that they felt "very safe" walking alone after dark, so the conditional relative frequency is
For female responsdents, we obtain the conditional relative frequency
Alternatively, we could condition on the event "respondent fills in the survey stating they feel very safe" and find the conditional relative frequency of being male or female. Here we work solely with the column that describes this response. So for relative frequency for males of
conditional on reporting they felt safe walking alone at night and for females
The question of which is the most appropriate conditional relative frequency depends on the question that is being asked of the data.
Bibliography
Blitzstein, Joseph K., and Jessica Hwang. Introduction to Probability. Boca Raton, FL: Chapman, 2015.
ESDS Government. Crime Survey for England and Wales, 2011-2012: Teaching Dataset. Manchester, UK: U Manchester, 2013.
Forbes, Catherine, et al. Statistical Distributions. Hoboken, NJ: Wiley, 2011.